Module PDESolver
PDESolver — A Spectral Method PDE Solver with Symbolic Capabilities
Overview
PDESolver is a powerful Python module for solving partial differential equations (PDEs) using spectral methods, enhanced with symbolic capabilities via SymPy. It combines high-precision numerical methods with symbolic analysis of differential and pseudo-differential operators, making it suitable for both research and educational applications.
Key Features
- Symbolic Parsing: Define PDEs using SymPy expressions for full symbolic manipulation
- Spectral Methods: Uses Fourier transforms for high-accuracy spatial differentiation
- Nonlinear Support: Handles nonlinear terms via pseudo-spectral evaluation and dealiasing
- Time Integration:
- Exponential time stepping for linear systems
- ETD-RK4 (Exponential Time Differencing with 4th-order Runge-Kutta) for stiff or nonlinear systems
- Pseudo-Differential Operators (
psiOp):- Define equations using
psiOp(symbol, u)for arbitrary pseudo-differential symbols - Supports symbolic inversion, adjoint computation, and asymptotic expansions
- Define equations using
- Boundary Conditions:
- Periodic (via FFT)
- Dirichlet (via pseudo-differential operator inversion)
- Interactive Analysis:
- Explore symbol properties (
|p(x, ξ)|, group velocity, wavefront sets) - Visualize Hamiltonian flows and characteristic sets
- Explore symbol properties (
- Visualization:
- Animate solutions in 1D/2D
- Plot real, imaginary, absolute, or phase components of complex solutions
Symbolic Workflow
PDESolver supports full symbolic definition of PDEs using SymPy syntax. It automatically extracts and analyzes:
- Linear operators in frequency space
L(k) - Dispersion relations
ω(k) - Nonlinear terms
- Pseudo-differential operators (
psiOp) - Source terms
Example Definition
from PDESolver import *
# Define PDE
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
#equation = Eq(diff(u, t, t), diff(u, x, 2) - u) # boundary_condition : 'periodic'
equation = Eq(diff(u(t,x), t), -psiOp(xi**2 + 1, u(t,x))) # boundary_condition : 'periodic' or 'dirichlet'
# Init solver
solver = PDESolver(equation)
# Setup domain
solver.setup(
Lx=2*np.pi, Nx=256,
Lt=2.0, Nt=1000,
initial_condition=lambda x: np.sin(x),
initial_velocity=lambda x: 0*x,
boundary_condition='periodic' # or 'dirichlet'
)
# Solve & animate
solver.solve()
ani = solver.animate(component='real')
HTML(ani.to_jshtml())
Numerical Methods
Spectral Differentiation
- Uses FFT-based spatial differentiation.
- Dealiasing is applied to nonlinear terms using a sharp cutoff.
- Handles 1D and 2D spatial domains.
Time Integration
- First-order evolution:
- Default exponential stepping.
- ETD-RK4 support for stiff or nonlinear systems.
- Second-order evolution:
- Leapfrog-style update.
- ETD-RK4 adapted for second-order systems.
- Supports acceleration from pseudo-differential operators.
Pseudo-Differential Operators
- Symbolic expressions like
xi**2 + 1define the operator symbol. - Evaluated using the Kohn–Nirenberg quantization.
- Supports non-periodic domains and Dirichlet boundary conditions through symbolic inversion.
Interactive Symbol Analysis
Use interactive_symbol_analysis(pseudo_op) to explore:
- Symbol amplitude and phase
- Group velocity fields
- Hamiltonian flows
- Wavefront sets
- Characteristic sets
- Micro-support estimates
This is particularly useful for studying:
- Wave propagation
- Singularity propagation
- Stability and dispersion properties
- Microlocal behavior of solutions
Example Use Case
from PDESolver import *
# Definition of symbols
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Evolution equation: ∂²u/∂t² = -ψOp(x² + ξ², u)
p_expr = x**2 + xi**2
equation = Eq(diff(u(t,x), t, t), -psiOp(p_expr, u(t,x)))
# Creation of the solver
solver = PDESolver(equation)
# Parameters
Lx = 12.0
Nx = 256
Lt = 3.0
Nt = 600
n = 2 # Order of Hermite
lambda_n = 2 * n + 1
# Initial function: u₀(x) = Hₙ(x) * exp(-x² / 2)
initial_condition = lambda x: eval_hermite(n, x) * np.exp(-x**2 / 2)
# Zero initial velocity: ∂ₜ u(0,x) = 0
initial_velocity = lambda x: 0.0 * x
# Exact solution
def u_exact(x, t):
return np.cos(np.sqrt(lambda_n) * t) * eval_hermite(n, x) * np.exp(-x**2 / 2)
# Solver setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition,
initial_velocity=initial_velocity,
)
# Solving
solver.solve()
# Validation tests
n_test = 5
for i in range(n_test + 1):
t_eval = i * Lt / n_test
solver.test(u_exact=u_exact, t_eval=t_eval, threshold=50, component='real')
Applications
PDESolver is ideal for:
- Educational tools (visualization of PDE solutions and symbolic analysis)
- Microlocal analysis (wavefront sets, Hamiltonian flows)
- Operator theory (pseudo-differential calculus, inversion, adjoints)
Dependencies
- numpy, scipy
- sympy for symbolic manipulation
- matplotlib for visualization
- ipywidgets for interactive analysis
- scipy.fft, scipy.integrate, scipy.signal
Expand source code
# Copyright 2025 Philippe Billet
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
PDESolver — A Spectral Method PDE Solver with Symbolic Capabilities
Overview
--------
PDESolver is a powerful Python module for solving partial differential equations (PDEs)
using spectral methods, enhanced with symbolic capabilities via SymPy. It combines high-precision
numerical methods with symbolic analysis of differential and pseudo-differential operators,
making it suitable for both research and educational applications.
Key Features
------------
- **Symbolic Parsing**: Define PDEs using SymPy expressions for full symbolic manipulation
- **Spectral Methods**: Uses Fourier transforms for high-accuracy spatial differentiation
- **Nonlinear Support**: Handles nonlinear terms via pseudo-spectral evaluation and dealiasing
- **Time Integration**:
- Exponential time stepping for linear systems
- ETD-RK4 (Exponential Time Differencing with 4th-order Runge-Kutta) for stiff or nonlinear systems
- **Pseudo-Differential Operators (`psiOp`)**:
- Define equations using `psiOp(symbol, u)` for arbitrary pseudo-differential symbols
- Supports symbolic inversion, adjoint computation, and asymptotic expansions
- **Boundary Conditions**:
- Periodic (via FFT)
- Dirichlet (via pseudo-differential operator inversion)
- **Interactive Analysis**:
- Explore symbol properties (`|p(x, ξ)|`, group velocity, wavefront sets)
- Visualize Hamiltonian flows and characteristic sets
- **Visualization**:
- Animate solutions in 1D/2D
- Plot real, imaginary, absolute, or phase components of complex solutions
Symbolic Workflow
-----------------
PDESolver supports full symbolic definition of PDEs using SymPy syntax. It automatically extracts and analyzes:
- Linear operators in frequency space `L(k)`
- Dispersion relations `ω(k)`
- Nonlinear terms
- Pseudo-differential operators (`psiOp`)
- Source terms
Example Definition
------------------
```python
from PDESolver import *
# Define PDE
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
#equation = Eq(diff(u, t, t), diff(u, x, 2) - u) # boundary_condition : 'periodic'
equation = Eq(diff(u(t,x), t), -psiOp(xi**2 + 1, u(t,x))) # boundary_condition : 'periodic' or 'dirichlet'
# Init solver
solver = PDESolver(equation)
# Setup domain
solver.setup(
Lx=2*np.pi, Nx=256,
Lt=2.0, Nt=1000,
initial_condition=lambda x: np.sin(x),
initial_velocity=lambda x: 0*x,
boundary_condition='periodic' # or 'dirichlet'
)
# Solve & animate
solver.solve()
ani = solver.animate(component='real')
HTML(ani.to_jshtml())
```
Numerical Methods
-----------------
Spectral Differentiation
-----------------
- Uses FFT-based spatial differentiation.
- Dealiasing is applied to nonlinear terms using a sharp cutoff.
- Handles 1D and 2D spatial domains.
Time Integration
-----------------
- First-order evolution:
- Default exponential stepping.
- ETD-RK4 support for stiff or nonlinear systems.
- Second-order evolution:
- Leapfrog-style update.
- ETD-RK4 adapted for second-order systems.
- Supports acceleration from pseudo-differential operators.
Pseudo-Differential Operators
-----------------------------
- Symbolic expressions like `xi**2 + 1` define the operator symbol.
- Evaluated using the Kohn–Nirenberg quantization.
- Supports non-periodic domains and Dirichlet boundary conditions through symbolic inversion.
Interactive Symbol Analysis
---------------------------
Use `interactive_symbol_analysis(pseudo_op)` to explore:
- Symbol amplitude and phase
- Group velocity fields
- Hamiltonian flows
- Wavefront sets
- Characteristic sets
- Micro-support estimates
This is particularly useful for studying:
- Wave propagation
- Singularity propagation
- Stability and dispersion properties
- Microlocal behavior of solutions
Example Use Case
----------------
```python
from PDESolver import *
# Definition of symbols
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Evolution equation: ∂²u/∂t² = -ψOp(x² + ξ², u)
p_expr = x**2 + xi**2
equation = Eq(diff(u(t,x), t, t), -psiOp(p_expr, u(t,x)))
# Creation of the solver
solver = PDESolver(equation)
# Parameters
Lx = 12.0
Nx = 256
Lt = 3.0
Nt = 600
n = 2 # Order of Hermite
lambda_n = 2 * n + 1
# Initial function: u₀(x) = Hₙ(x) * exp(-x² / 2)
initial_condition = lambda x: eval_hermite(n, x) * np.exp(-x**2 / 2)
# Zero initial velocity: ∂ₜ u(0,x) = 0
initial_velocity = lambda x: 0.0 * x
# Exact solution
def u_exact(x, t):
return np.cos(np.sqrt(lambda_n) * t) * eval_hermite(n, x) * np.exp(-x**2 / 2)
# Solver setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition,
initial_velocity=initial_velocity,
)
# Solving
solver.solve()
# Validation tests
n_test = 5
for i in range(n_test + 1):
t_eval = i * Lt / n_test
solver.test(u_exact=u_exact, t_eval=t_eval, threshold=50, component='real')
```
Applications
------------
PDESolver is ideal for:
- Educational tools (visualization of PDE solutions and symbolic analysis)
- Microlocal analysis (wavefront sets, Hamiltonian flows)
- Operator theory (pseudo-differential calculus, inversion, adjoints)
Dependencies
------------
- numpy, scipy
- sympy for symbolic manipulation
- matplotlib for visualization
- ipywidgets for interactive analysis
- scipy.fft, scipy.integrate, scipy.signal
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import fft2, ifft2, fft, ifft, fftfreq, fftshift, ifftshift
from scipy.signal.windows import hann
from scipy.integrate import solve_ivp
from scipy.ndimage import maximum_filter
from sympy import (
symbols, Function,
solve, pprint, Mul,
lambdify, expand, Eq, simplify, trigsimp, N,
radsimp, ratsimp, cancel,
Lambda, Piecewise, Basic, degree, Pow, preorder_traversal,
powdenest, expand,
sqrt, I, pi, series, oo,
re, im, arg, Abs, conjugate,
sin, cos, tan, cot, sec, csc, sinc,
asin, acos, atan, acot, asec, acsc,
sinh, cosh, tanh, coth, sech, csch,
asinh, acosh, atanh, acoth, asech, acsch,
exp, ln, log, factorial,
diff, Derivative, integrate,
fourier_transform, inverse_fourier_transform,
)
from sympy.core.function import AppliedUndef
from scipy.special import legendre, eval_hermite, airy, eval_genlaguerre
from matplotlib import cm
from matplotlib.animation import FuncAnimation
import matplotlib.animation as animation
from matplotlib import rc
from functools import partial
from misc import *
from IPython.display import display, clear_output, HTML
from ipywidgets import interact, FloatSlider, Dropdown, widgets
from IPython.display import display, clear_output
from itertools import product
import os
plt.rcParams['text.usetex'] = False
FFT_WORKERS = max(1, os.cpu_count() // 2)
class Op(Function):
"""Custom symbolic wrapper for pseudo-differential operators in Fourier space.
Usage: Op(symbol_expr, u)
"""
nargs = 2
class psiOp(Function):
"""Symbolic wrapper for PseudoDifferentialOperator.
Usage: psiOp(symbol_expr, u)
"""
nargs = 2 # (expr, u)
class PseudoDifferentialOperator:
"""
Pseudo-differential operator with dynamic symbol evaluation on spatial grids.
Supports both 1D and 2D operators, and can be defined explicitly (symbol mode)
or extracted automatically from symbolic equations (auto mode).
Parameters
----------
expr : sympy expression
Symbolic expression representing the pseudo-differential symbol.
vars_x : list of sympy symbols
Spatial variables (e.g., [x] for 1D, [x, y] for 2D).
var_u : sympy function, optional
Function u(x, t) used in auto mode to extract the operator symbol.
mode : str, {'symbol', 'auto'}
- 'symbol': directly uses expr as the operator symbol.
- 'auto': computes the symbol automatically by applying expr to exp(i x ξ).
Attributes
----------
dim : int
Spatial dimension (1 or 2).
fft, ifft : callable
Fast Fourier transform and inverse (scipy.fft or scipy.fft2).
p_func : callable
Evaluated symbol function ready for numerical use.
Notes
-----
- In 'symbol' mode, `expr` should be expressed in terms of spatial variables and frequency variables (ξ, η).
- In 'auto' mode, the symbol is derived by applying the differential expression to a complex exponential.
- Frequency variables are internally named 'xi' and 'eta' for consistency.
- Uses numpy for numerical evaluation and scipy.fft for FFT operations.
Examples
--------
>>> # Example 1: 1D Laplacian operator (symbol mode)
>>> from sympy import symbols
>>> x, xi = symbols('x xi', real=True)
>>> op = PseudoDifferentialOperator(expr=xi**2, vars_x=[x], mode='symbol')
>>> # Example 2: 1D transport operator (auto mode)
>>> from sympy import Function
>>> u = Function('u')
>>> expr = u(x).diff(x)
>>> op = PseudoDifferentialOperator(expr=expr, vars_x=[x], var_u=u(x), mode='auto')
"""
def __init__(self, expr, vars_x, var_u=None, mode='symbol'):
self.dim = len(vars_x)
self.mode = mode
self.symbol_cached = None
self.expr = expr
self.vars_x = vars_x
if self.dim == 1:
x, = vars_x
xi_internal = symbols('xi', real=True)
expr = expr.subs(symbols('xi', real=True), xi_internal)
self.fft = partial(fft, workers=FFT_WORKERS)
self.ifft = partial(ifft, workers=FFT_WORKERS)
if mode == 'symbol':
self.p_func = lambdify((x, xi_internal), expr, 'numpy')
self.symbol = expr
elif mode == 'auto':
if var_u is None:
raise ValueError("var_u must be provided in mode='auto'")
exp_i = exp(I * x * xi_internal)
P_ei = expr.subs(var_u, exp_i)
symbol = simplify(P_ei / exp_i)
symbol = expand(symbol)
self.symbol = symbol
self.p_func = lambdify((x, xi_internal), symbol, 'numpy')
else:
raise ValueError("mode must be 'auto' or 'symbol'")
elif self.dim == 2:
x, y = vars_x
xi_internal, eta_internal = symbols('xi eta', real=True)
expr = expr.subs(symbols('xi', real=True), xi_internal)
expr = expr.subs(symbols('eta', real=True), eta_internal)
self.fft = partial(fft2, workers=FFT_WORKERS)
self.ifft = partial(ifft2, workers=FFT_WORKERS)
if mode == 'symbol':
self.symbol = expr
self.p_func = lambdify((x, y, xi_internal, eta_internal), expr, 'numpy')
elif mode == 'auto':
if var_u is None:
raise ValueError("var_u must be provided in mode='auto'")
exp_i = exp(I * (x * xi_internal + y * eta_internal))
P_ei = expr.subs(var_u, exp_i)
symbol = simplify(P_ei / exp_i)
symbol = expand(symbol)
self.symbol = symbol
self.p_func = lambdify((x, y, xi_internal, eta_internal), symbol, 'numpy')
else:
raise ValueError("mode must be 'auto' or 'symbol'")
else:
raise NotImplementedError("Only 1D and 2D supported")
print("\nsymbol = ")
pprint(expr, num_columns=150)
def evaluate(self, X, Y, KX, KY, cache=True):
"""
Evaluate the pseudo-differential operator's symbol on a grid of spatial and frequency coordinates.
The method dynamically selects between 1D and 2D evaluation based on the spatial dimension.
If caching is enabled and a cached symbol exists, it returns the cached result to avoid recomputation.
Parameters
----------
X, Y : ndarray
Spatial grid coordinates. In 1D, Y is ignored.
KX, KY : ndarray
Frequency grid coordinates. In 1D, KY is ignored.
cache : bool, default=True
If True, stores the computed symbol for reuse in subsequent calls to avoid redundant computation.
Returns
-------
ndarray
Evaluated symbol values over the input grid. Shape matches the input spatial/frequency grids.
Raises
------
NotImplementedError
If the spatial dimension is not 1D or 2D.
"""
if cache and self.symbol_cached is not None:
return self.symbol_cached
if self.dim == 1:
symbol = self.p_func(X, KX)
elif self.dim == 2:
symbol = self.p_func(X, Y, KX, KY)
if cache:
self.symbol_cached = symbol
return symbol
def clear_cache(self):
"""
Clear cached symbol evaluations.
"""
self.symbol_cached = None
def principal_symbol(self, order=1):
"""
Compute the leading homogeneous component of the pseudo-differential symbol.
This method extracts the principal part of the symbol, which is the dominant
term under high-frequency asymptotics (|ξ| → ∞). The expansion is performed
in polar coordinates for 2D symbols to maintain rotational symmetry, then
converted back to Cartesian form.
Parameters
----------
order : int
Order of the asymptotic expansion in powers of 1/ρ, where ρ = |ξ| in 1D
or ρ = sqrt(ξ² + η²) in 2D. Only the leading-order term is returned.
Returns
-------
sympy.Expr
The principal symbol component, homogeneous of degree `m - order`, where
`m` is the original symbol's order.
Notes:
- In 1D, uses direct series expansion in ξ.
- In 2D, expands in radial variable ρ while preserving angular dependence.
- Useful for microlocal analysis and constructing parametrices.
"""
p = self.expr
if self.dim == 1:
xi = symbols('xi', real=True, positive=True)
return simplify(series(p, xi, oo, n=order).removeO())
elif self.dim == 2:
xi, eta = symbols('xi eta', real=True, positive=True)
# Homogeneous radial expansion: we set (ξ, η) = ρ (cosθ, sinθ)
rho, theta = symbols('rho theta', real=True, positive=True)
p_rho = p.subs({xi: rho * cos(theta), eta: rho * sin(theta)})
expansion = series(p_rho, rho, oo, n=order).removeO()
# Revert back to (ξ, η)
expansion_cart = expansion.subs({rho: sqrt(xi**2 + eta**2),
cos(theta): xi / sqrt(xi**2 + eta**2),
sin(theta): eta / sqrt(xi**2 + eta**2)})
return simplify(powdenest(expansion_cart, force=True))
def is_homogeneous(self, tol=1e-10):
"""
Check whether the symbol is homogeneous in the frequency variables.
Returns
-------
(bool, Rational or float or None)
Tuple (is_homogeneous, degree) where:
- is_homogeneous: True if the symbol satisfies p(λξ, λη) = λ^m * p(ξ, η)
- degree: the detected degree m if homogeneous, or None
"""
from sympy import symbols, simplify, expand, Eq
from sympy.abc import l
if self.dim == 1:
xi = symbols('xi', real=True, positive=True)
l = symbols('l', real=True, positive=True)
p = self.expr
p_scaled = p.subs(xi, l * xi)
ratio = simplify(p_scaled / p)
if ratio.has(xi):
return False, None
try:
deg = simplify(ratio).as_base_exp()[1]
return True, deg
except Exception:
return False, None
elif self.dim == 2:
xi, eta = symbols('xi eta', real=True, positive=True)
l = symbols('l', real=True, positive=True)
p = self.expr
p_scaled = p.subs({xi: l * xi, eta: l * eta})
ratio = simplify(p_scaled / p)
# If ratio == l**m with no (xi, eta) left, it's homogeneous
if ratio.has(xi, eta):
return False, None
try:
base, exp = ratio.as_base_exp()
if base == l:
return True, exp
except Exception:
pass
return False, None
def symbol_order(self, max_order=10, tol=1e-3):
"""
Estimate the homogeneity order of the pseudo-differential symbol in high-frequency asymptotics.
This method attempts to determine the leading-order behavior of the symbol p(x, ξ) or p(x, y, ξ, η)
as |ξ| → ∞ (in 1D) or |(ξ, η)| → ∞ (in 2D). The returned value represents the asymptotic growth or decay rate,
which is essential for understanding the regularity and mapping properties of the corresponding operator.
The function uses symbolic preprocessing to ensure proper factorization of frequency variables,
especially in sqrt and power expressions, to avoid erroneous order detection (e.g., due to hidden scaling).
Parameters
----------
max_order : int, optional
Maximum number of terms to consider in the series expansion. Default is 10.
tol : float, optional
Tolerance threshold for evaluating the coefficient magnitude. If the coefficient is too small,
the detected order may be discarded. Default is 1e-3.
Returns
-------
float or None
- If the symbol is homogeneous, returns its exact homogeneity degree as a float.
- Otherwise, estimates the dominant asymptotic order from leading terms in the expansion.
- Returns None if no valid order could be determined.
Notes
-----
- In 1D:
Two strategies are used:
1. Expand directly in xi at infinity.
2. Substitute xi = 1/z and expand around z = 0.
- In 2D:
- Transform the symbol into polar coordinates: (xi, eta) = rho*(cos(theta), sin(theta)).
- Expand in rho at infinity, then extract the leading term's power.
- An alternative substitution using 1/z is also tried if the first method fails.
- Preprocessing steps:
- Sqrt expressions involving frequencies are rewritten to isolate the leading variable.
- Power expressions are factored explicitly to ensure correct symbolic scaling.
- If the symbol is not homogeneous, a warning is issued, and the result should be interpreted with care.
- For non-homogeneous symbols, only the principal asymptotic term is considered.
Raises
------
NotImplementedError
If the spatial dimension is neither 1 nor 2.
"""
from sympy import (
symbols, series, simplify, sqrt, cos, sin, oo, powdenest, radsimp,
expand, expand_power_base
)
def preprocess_sqrt(expr, freq):
return expr.replace(
lambda e: e.func == sqrt and freq in e.free_symbols,
lambda e: freq * sqrt(1 + (e.args[0] - freq**2) / freq**2)
)
def preprocess_power(expr, freq):
return expr.replace(
lambda e: e.is_Pow and freq in e.free_symbols,
lambda e: freq**e.exp * (1 + e.base / freq**e.base.as_powers_dict().get(freq, 0))**e.exp
)
def validate_order(power, coeff, vars_x, tol):
if power is None:
return None
if any(v in coeff.free_symbols for v in vars_x):
print("⚠️ Coefficient depends on spatial variables; ignoring")
return None
try:
coeff_val = abs(float(coeff.evalf()))
if coeff_val < tol:
print(f"⚠️ Coefficient too small ({coeff_val:.2e} < {tol})")
return None
except Exception as e:
print(f"⚠️ Coefficient evaluation failed: {e}")
return None
return int(power) if power == int(power) else float(power)
# Homogeneity check
is_homog, degree = self.is_homogeneous()
if is_homog:
return float(degree)
else:
print("⚠️ The symbol is not homogeneous. The asymptotic order is not well defined.")
if self.dim == 1:
x = self.vars_x[0]
xi = symbols('xi', real=True, positive=True)
try:
print("1D symbol_order - method 1")
expr = preprocess_sqrt(self.expr, xi)
s = series(expr, xi, oo, n=max_order).removeO()
lead = simplify(powdenest(s.as_leading_term(xi), force=True))
power = lead.as_powers_dict().get(xi, None)
coeff = lead / xi**power if power is not None else 0
print("lead =", lead)
print("power =", power)
print("coeff =", coeff)
order = validate_order(power, coeff, [x], tol)
if order is not None:
return order
except Exception:
pass
try:
print("1D symbol_order - method 2")
z = symbols('z', real=True, positive=True)
expr_z = preprocess_sqrt(self.expr.subs(xi, 1/z), 1/z)
s = series(expr_z, z, 0, n=max_order).removeO()
lead = simplify(powdenest(s.as_leading_term(z), force=True))
power = lead.as_powers_dict().get(z, None)
coeff = lead / z**power if power is not None else 0
print("lead =", lead)
print("power =", power)
print("coeff =", coeff)
order = validate_order(power, coeff, [x], tol)
if order is not None:
return -order
except Exception as e:
print(f"⚠️ fallback z failed: {e}")
return None
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True, positive=True)
rho, theta = symbols('rho theta', real=True, positive=True)
try:
print("2D symbol_order - method 1")
p_rho = self.expr.subs({xi: rho * cos(theta), eta: rho * sin(theta)})
p_rho = preprocess_power(preprocess_sqrt(p_rho, rho), rho)
s = series(simplify(p_rho), rho, oo, n=max_order).removeO()
lead = radsimp(simplify(powdenest(s.as_leading_term(rho), force=True)))
power = lead.as_powers_dict().get(rho, None)
coeff = lead / rho**power if power is not None else 0
print("lead =", lead)
print("power =", power)
print("coeff =", coeff)
order = validate_order(power, coeff, [x, y], tol)
if order is not None:
return order
except Exception as e:
print(f"⚠️ polar expansion failed: {e}")
try:
print("2D symbol_order - method 2")
z = symbols('z', real=True, positive=True)
xi_eta = {xi: (1/z) * cos(theta), eta: (1/z) * sin(theta)}
p_rho = preprocess_sqrt(self.expr.subs(xi_eta), 1/z)
s = series(simplify(p_rho), z, 0, n=max_order).removeO()
lead = radsimp(simplify(powdenest(s.as_leading_term(z), force=True)))
power = lead.as_powers_dict().get(z, None)
coeff = lead / z**power if power is not None else 0
print("lead =", lead)
print("power =", power)
print("coeff =", coeff)
order = validate_order(power, coeff, [x, y], tol)
if order is not None:
return -order
except Exception as e:
print(f"⚠️ fallback z (2D) failed: {e}")
return None
else:
raise NotImplementedError("Only 1D and 2D supported.")
def asymptotic_expansion(self, order=3):
"""
Compute the asymptotic expansion of the symbol as |ξ| → ∞ (high-frequency regime).
This method expands the pseudo-differential symbol in inverse powers of the
frequency variable(s), either in 1D or 2D. It handles both polynomial and
exponential symbols by performing a series expansion in 1/|ξ| up to the specified order.
The expansion is performed directly in Cartesian coordinates for 1D symbols.
For 2D symbols, the method uses polar coordinates (ρ, θ) to perform the expansion
at infinity in ρ, then converts the result back to Cartesian coordinates.
Parameters
----------
order : int, optional
Maximum order of the asymptotic expansion. Default is 3.
Returns
-------
sympy.Expr
The asymptotic expansion of the symbol up to the given order, expressed in Cartesian coordinates.
If expansion fails, returns the original unexpanded symbol.
Notes:
- In 1D: expansion is performed directly in terms of ξ.
- In 2D: the symbol is first rewritten in polar coordinates (ρ,θ), expanded asymptotically
in ρ → ∞, then converted back to Cartesian coordinates (ξ,η).
- Handles special case when the symbol is an exponential function by expanding its argument.
- Symbolic normalization is applied early (via `simplify`) for 2D expressions to improve convergence.
- Robust to failures: catches exceptions and issues warnings instead of raising errors.
- Final expression is simplified using `powdenest` and `expand` for improved readability.
"""
p = self.expr
if self.dim == 1:
xi = symbols('xi', real=True, positive=True)
try:
# Case: exponential function
if p.func == exp and len(p.args) == 1:
arg = p.args[0]
arg_series = series(arg, xi, oo, n=order).removeO()
expanded = series(exp(expand(arg_series)), xi, oo, n=order).removeO()
return simplify(powdenest(expanded, force=True))
else:
expanded = series(p, xi, oo, n=order).removeO()
return simplify(powdenest(expanded, force=True))
except Exception as e:
print(f"Warning: 1D expansion failed: {e}")
return p
elif self.dim == 2:
xi, eta = symbols('xi eta', real=True, positive=True)
rho, theta = symbols('rho theta', real=True, positive=True)
# Normalize before substitution
p = simplify(p)
# Substitute polar coordinates
p_polar = p.subs({
xi: rho * cos(theta),
eta: rho * sin(theta)
})
try:
# Handle exponentials
if p_polar.func == exp and len(p_polar.args) == 1:
arg = p_polar.args[0]
arg_series = series(arg, rho, oo, n=order).removeO()
expanded = series(exp(expand(arg_series)), rho, oo, n=order).removeO()
else:
expanded = series(p_polar, rho, oo, n=order).removeO()
# Convert back to Cartesian
norm = sqrt(xi**2 + eta**2)
expansion_cart = expanded.subs({
rho: norm,
cos(theta): xi / norm,
sin(theta): eta / norm
})
# Final simplifications
result = simplify(powdenest(expansion_cart, force=True))
result = expand(result)
return result
except Exception as e:
print(f"Warning: 2D expansion failed: {e}")
return p
def compose_asymptotic(self, other, order=1):
"""
Compose this pseudo-differential operator with another using formal asymptotic expansion.
This method computes the composition symbol via an asymptotic expansion in powers of
derivatives, following the symbolic calculus of pseudo-differential operators. The
composition is performed up to the specified order and respects the dimensionality
(1D or 2D) of the operators.
Parameters
----------
other : PseudoDifferentialOperator
The pseudo-differential operator to compose with this one.
order : int, default=1
Maximum order of the asymptotic expansion. Higher values include more terms in the
symbolic composition, increasing accuracy at the cost of complexity.
Returns
-------
sympy.Expr
Symbolic expression representing the asymptotic expansion of the composed operator.
Notes
-----
- In 1D, the composition uses the formula:
(p ∘ q)(x, ξ) ~ Σₙ (1/n!) ∂_ξⁿ p(x, ξ) ∂_xⁿ q(x, ξ) (i)^{-n}
- In 2D, the multi-index generalization is used:
(p ∘ q)(x, y, ξ, η) ~ Σₙ Σᵢ (1/(i! j!)) ∂_ξⁱ∂_ηʲ p ∂_xⁱ∂_yʲ q (i)^{-n}, where n = i + j.
- This expansion is valid for symbols admitting an asymptotic series representation.
- Operators must be defined on the same spatial domain (same dimension).
"""
assert self.dim == other.dim, "Operator dimensions must match"
p, q = self.expr, other.expr
if self.dim == 1:
x = self.vars_x[0]
xi = symbols('xi', real=True)
result = 0
for n in range(order + 1):
term = (1 / factorial(n)) * diff(p, xi, n) * diff(q, x, n) * (1j)**(-n)
result += term
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
result = 0
for n in range(order + 1):
for i in range(n + 1):
j = n - i
term = (1 / (factorial(i) * factorial(j))) * \
diff(p, xi, i, eta, j) * diff(q, x, i, y, j) * (1j)**(-n)
result += term
return result
def right_inverse_asymptotic(self, order=1):
"""
Construct a formal right inverse R of the pseudo-differential operator P such that
the composition P ∘ R equals the identity plus a smoothing operator of order -order.
This method computes an asymptotic expansion for the right inverse using recursive
corrections based on derivatives of the symbol p(x, ξ) and lower-order terms of R.
Parameters
----------
order : int
Number of terms to include in the asymptotic expansion. Higher values improve
approximation at the cost of complexity and computational effort.
Returns
-------
sympy.Expr
The symbolic expression representing the formal right inverse R(x, ξ), which satisfies:
P ∘ R = Id + O(⟨ξ⟩^{-order}), where ⟨ξ⟩ = (1 + |ξ|²)^{1/2}.
Notes
-----
- In 1D: The recursion involves spatial derivatives of R and derivatives of p with respect to ξ.
- In 2D: The multi-index generalization is used with mixed derivatives in ξ and η.
- The construction relies on the non-vanishing of the principal symbol p to ensure invertibility.
- Each term in the expansion corresponds to higher-order corrections involving commutators
between the operator P and the current approximation of R.
"""
p = self.expr
if self.dim == 1:
x = self.vars_x[0]
xi = symbols('xi', real=True)
r = 1 / p.subs(xi, xi) # r0
R = r
for n in range(1, order + 1):
term = 0
for k in range(1, n + 1):
coeff = (1j)**(-k) / factorial(k)
inner = diff(p, xi, k) * diff(R, x, k)
term += coeff * inner
R = R - r * term
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
r = 1 / p.subs({xi: xi, eta: eta})
R = r
for n in range(1, order + 1):
term = 0
for k1 in range(n + 1):
for k2 in range(n + 1 - k1):
if k1 + k2 == 0: continue
coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2))
dp = diff(p, xi, k1, eta, k2)
dR = diff(R, x, k1, y, k2)
term += coeff * dp * dR
R = R - r * term
return R
def left_inverse_asymptotic(self, order=1):
"""
Construct a formal left inverse L such that the composition L ∘ P equals the identity
operator up to terms of order ξ^{-order}. This expansion is performed asymptotically
at infinity in the frequency variable(s).
The left inverse is built iteratively using symbolic differentiation and the
method of asymptotic expansions for pseudo-differential operators. It ensures that:
L(P(x,ξ),x,D) ∘ P(x,D) = Id + smoothing operator of order -order
Parameters
----------
order : int, optional
Maximum number of terms in the asymptotic expansion (default is 1). Higher values
yield more accurate inverses at the cost of increased computational complexity.
Returns
-------
sympy.Expr
Symbolic expression representing the principal symbol of the formal left inverse
operator L(x,ξ). This expression depends on spatial variables and frequencies,
and includes correction terms up to the specified order.
Notes
-----
- In 1D: Uses recursive application of the Leibniz formula for symbols.
- In 2D: Generalizes to multi-indices for mixed derivatives in (x,y) and (ξ,η).
- Each term involves combinations of derivatives of the original symbol p(x,ξ) and
previously computed terms of the inverse.
- Coefficients include powers of 1j (i) and factorial normalization for derivative terms.
"""
p = self.expr
if self.dim == 1:
x = self.vars_x[0]
xi = symbols('xi', real=True)
l = 1 / p.subs(xi, xi)
L = l
for n in range(1, order + 1):
term = 0
for k in range(1, n + 1):
coeff = (1j)**(-k) / factorial(k)
inner = diff(L, xi, k) * diff(p, x, k)
term += coeff * inner
L = L - term * l
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
l = 1 / p.subs({xi: xi, eta: eta})
L = l
for n in range(1, order + 1):
term = 0
for k1 in range(n + 1):
for k2 in range(n + 1 - k1):
if k1 + k2 == 0: continue
coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2))
dp = diff(p, x, k1, y, k2)
dL = diff(L, xi, k1, eta, k2)
term += coeff * dL * dp
L = L - term * l
return L
def formal_adjoint(self):
"""
Compute the formal adjoint symbol P* of the pseudo-differential operator.
The adjoint is defined such that for any test functions u and v,
⟨P u, v⟩ = ⟨u, P* v⟩ holds in the distributional sense. This is obtained by
taking the complex conjugate of the symbol and expanding it asymptotically
at infinity to ensure proper behavior under integration by parts.
Returns
-------
sympy.Expr
The adjoint symbol P*(x, ξ) in 1D or P*(x, y, ξ, η) in 2D.
Notes:
- In 1D, the expansion is performed in powers of 1/|ξ|.
- In 2D, the expansion is radial in |ξ| = sqrt(ξ² + η²).
- This method ensures symbolic simplifications for readability and efficiency.
"""
p = self.expr
if self.dim == 1:
x, = self.vars_x
xi = symbols('xi', real=True)
p_star = conjugate(p)
p_star = simplify(series(p_star, xi, oo, n=6).removeO())
return p_star
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
p_star = conjugate(p)
p_star = simplify(series(p_star, sqrt(xi**2 + eta**2), oo, n=6).removeO())
return p_star
def symplectic_flow(self):
"""
Compute the Hamiltonian vector field associated with the principal symbol.
This method derives the canonical equations of motion for the phase space variables
(x, ξ) in 1D or (x, y, ξ, η) in 2D, based on the Hamiltonian formalism. These describe
how position and frequency variables evolve under the flow generated by the symbol.
Returns
-------
dict
A dictionary containing the components of the Hamiltonian vector field:
- In 1D: keys are 'dx/dt' and 'dxi/dt', corresponding to dx/dt = ∂p/∂ξ and dξ/dt = -∂p/∂x.
- In 2D: keys are 'dx/dt', 'dy/dt', 'dxi/dt', and 'deta/dt', with similar definitions:
dx/dt = ∂p/∂ξ, dy/dt = ∂p/∂η, dξ/dt = -∂p/∂x, dη/dt = -∂p/∂y.
Notes
-----
- The Hamiltonian here is the principal symbol p(x, ξ) itself.
- This flow preserves the symplectic structure of phase space.
"""
if self.dim == 1:
x, = self.vars_x
xi = symbols('xi', real=True)
print("x = ", x)
print("xi = ", xi)
return {
'dx/dt': diff(self.symbol, xi),
'dxi/dt': -diff(self.symbol, x)
}
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
return {
'dx/dt': diff(self.symbol, xi),
'dy/dt': diff(self.symbol, eta),
'dxi/dt': -diff(self.symbol, x),
'deta/dt': -diff(self.symbol, y)
}
def is_elliptic_numerically(self, x_grid, xi_grid, threshold=1e-8):
"""
Check if the pseudo-differential symbol p(x, ξ) is elliptic over a given grid.
A symbol is considered elliptic if its magnitude |p(x, ξ)| remains bounded away from zero
across all points in the spatial-frequency domain. This method evaluates the symbol on a
grid of spatial and frequency coordinates and checks whether its minimum absolute value
exceeds a specified threshold.
Resampling is applied to large grids to prevent excessive memory usage, particularly in 2D.
Parameters
----------
x_grid : ndarray
Spatial grid: either a 1D array (x) or a tuple of two 1D arrays (x, y).
xi_grid : ndarray
Frequency grid: either a 1D array (ξ) or a tuple of two 1D arrays (ξ, η).
threshold : float, optional
Minimum acceptable value for |p(x, ξ)|. If the smallest evaluated symbol value falls below this,
the symbol is not considered elliptic.
Returns
-------
bool
True if the symbol is elliptic on the resampled grid, False otherwise.
"""
RESAMPLE_SIZE = 32 # Reduced size to prevent memory explosion
if self.dim == 1:
x_vals = x_grid
xi_vals = xi_grid
# Resampling if necessary
if len(x_vals) > RESAMPLE_SIZE:
x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE)
if len(xi_vals) > RESAMPLE_SIZE:
xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE)
X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij')
symbol_vals = self.p_func(X, XI)
elif self.dim == 2:
x_vals, y_vals = x_grid
xi_vals, eta_vals = xi_grid
# Spatial resampling
if len(x_vals) > RESAMPLE_SIZE:
x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE)
if len(y_vals) > RESAMPLE_SIZE:
y_vals = np.linspace(y_vals.min(), y_vals.max(), RESAMPLE_SIZE)
# Frequency resampling
if len(xi_vals) > RESAMPLE_SIZE:
xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE)
if len(eta_vals) > RESAMPLE_SIZE:
eta_vals = np.linspace(eta_vals.min(), eta_vals.max(), RESAMPLE_SIZE)
X, Y, XI, ETA = np.meshgrid(x_vals, y_vals, xi_vals, eta_vals, indexing='ij')
symbol_vals = self.p_func(X, Y, XI, ETA)
min_abs_val = np.min(np.abs(symbol_vals))
return min_abs_val > threshold
def is_self_adjoint(self, tol=1e-10):
"""
Check whether the pseudo-differential operator is formally self-adjoint (Hermitian).
A self-adjoint operator satisfies P = P*, where P* is the formal adjoint of P.
This property is essential for ensuring real-valued eigenvalues and stable evolution
in quantum mechanics and symmetric wave propagation.
Parameters
----------
tol : float
Tolerance for symbolic comparison between P and P*. Small numerical differences
below this threshold are considered equal.
Returns
-------
bool
True if the symbol p(x, ξ) equals its formal adjoint p*(x, ξ) within the given tolerance,
indicating that the operator is self-adjoint.
Notes:
- The formal adjoint is computed via conjugation and asymptotic expansion at infinity in ξ.
- Symbolic simplification is used to verify equality, ensuring robustness against superficial
expression differences.
"""
p = self.expr
p_star = self.formal_adjoint()
return simplify(p - p_star).equals(0)
def visualize_wavefront(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0):
"""
Visualize the wavefront set of the pseudo-differential operator's symbol by plotting |p(x, ξ)| or |p(x, y, ξ₀, η₀)|.
The wavefront set captures the location and direction of singularities in the symbol p, which is crucial in
microlocal analysis for understanding propagation of singularities and wave behavior.
In 1D, this method displays |p(x, ξ)| over the phase space (x, ξ). In 2D, it fixes the frequency values (ξ₀, η₀)
and visualizes |p(x, y, ξ₀, η₀)| over the spatial domain to highlight where the symbol interacts with the fixed frequency.
Parameters
----------
x_grid : ndarray
1D array of spatial coordinates (x) used for evaluation.
xi_grid : ndarray
1D array of frequency coordinates (ξ) used for visualization in 1D or slicing in 2D.
y_grid : ndarray, optional
1D array of second spatial coordinate (y), used only in 2D.
eta_grid : ndarray, optional
1D array of second frequency coordinate (η), not directly used but kept for API consistency.
xi0 : float, default=0.0
Fixed value of ξ for slicing in 2D visualization.
eta0 : float, default=0.0
Fixed value of η for slicing in 2D visualization.
Notes
-----
- In 1D:
Displays a 2D color map of |p(x, ξ)| over the phase space with axes (x, ξ).
- In 2D:
Evaluates and plots |p(x, y, ξ₀, η₀)| at fixed frequencies (ξ₀, η₀) over the spatial grid (x, y).
- Uses `imshow` for fast and scalable visualization with automatic aspect ratio adjustment.
- A colormap ('viridis') is used to enhance contrast and readability of the magnitude variations.
"""
if self.dim == 1:
X, XI = np.meshgrid(x_grid, xi_grid)
symbol_vals = self.p_func(X, XI)
plt.imshow(np.abs(symbol_vals), extent=[xi_grid.min(), xi_grid.max(), x_grid.min(), x_grid.max()],
aspect='auto', origin='lower', cmap='viridis')
elif self.dim == 2:
X, Y = np.meshgrid(x_grid, y_grid)
XI = np.full_like(X, xi0)
ETA = np.full_like(Y, eta0)
symbol_vals = self.p_func(X, Y, XI, ETA)
plt.imshow(np.abs(symbol_vals), extent=[x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()],
aspect='auto', origin='lower', cmap='viridis')
plt.colorbar(label='|Symbol|')
plt.xlabel('ξ/Position')
plt.ylabel('η/Position')
plt.title('Wavefront Set')
plt.show()
def visualize_fiber(self, x_grid, xi_grid, y0=0.0, x0=0.0):
"""
Plot the cotangent fiber structure at a fixed spatial point (x₀[, y₀]).
This visualization shows how the symbol p(x, ξ) behaves on the cotangent fiber
above a fixed spatial point. In microlocal analysis, this provides insight into
the frequency content of the operator at that location.
Parameters
----------
x_grid : ndarray
Spatial grid values (1D) for evaluation in 1D case.
xi_grid : ndarray
Frequency grid values (1D) for evaluation in both 1D and 2D cases.
x0 : float, optional
Fixed x-coordinate of the base point in space (1D or 2D).
y0 : float, optional
Fixed y-coordinate of the base point in space (2D only).
Notes
-----
- In 1D: Displays |p(x, ξ)| over the (x, ξ) phase plane near the fixed point.
- In 2D: Fixes (x₀, y₀) and evaluates p(x₀, y₀, ξ, η), showing the fiber over that point.
- The color map represents the magnitude of the symbol, highlighting regions where it vanishes or becomes singular.
Raises
------
NotImplementedError
If called in 2D with missing or improperly formatted grids.
"""
if self.dim == 1:
X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij')
symbol_vals = self.p_func(X, XI)
plt.contourf(X, XI, np.abs(symbol_vals), levels=50, cmap='viridis')
plt.colorbar(label='|Symbol|')
plt.xlabel('x (position)')
plt.ylabel('ξ (frequency)')
plt.title('Cotangent Fiber Structure')
plt.show()
elif self.dim == 2:
xi_grid2, eta_grid2 = np.meshgrid(xi_grid, xi_grid)
symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2)
plt.contourf(xi_grid, xi_grid, np.abs(symbol_vals), levels=50, cmap='viridis')
plt.colorbar(label='|Symbol|')
plt.xlabel('ξ')
plt.ylabel('η')
plt.title(f'Cotangent Fiber at x={x0}, y={y0}')
plt.show()
def visualize_symbol_amplitude(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0):
"""
Display the modulus |p(x, ξ)| or |p(x, y, ξ₀, η₀)| as a color map.
This method visualizes the amplitude of the pseudodifferential operator's symbol
in either 1D or 2D spatial configuration. In 2D, the frequency variables are fixed
to specified values (ξ₀, η₀) for visualization purposes.
Parameters
----------
x_grid, y_grid : ndarray
Spatial grids over which to evaluate the symbol. y_grid is optional and used only in 2D.
xi_grid, eta_grid : ndarray
Frequency grids. In 2D, these define the domain over which the symbol is evaluated,
but the visualization fixes ξ = ξ₀ and η = η₀.
xi0, eta0 : float, optional
Fixed frequency values for slicing in 2D visualization. Defaults to zero.
Notes
-----
- In 1D: Visualizes |p(x, ξ)| over the (x, ξ) grid.
- In 2D: Visualizes |p(x, y, ξ₀, η₀)| at fixed frequencies ξ₀ and η₀.
- The color intensity represents the magnitude of the symbol, highlighting regions where the symbol is large or small.
"""
if self.dim == 1:
X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij')
symbol_vals = self.p_func(X, XI)
plt.pcolormesh(X, XI, np.abs(symbol_vals), shading='auto')
plt.colorbar(label='|Symbol|')
plt.xlabel('x')
plt.ylabel('ξ')
plt.title('Symbol Amplitude |p(x, ξ)|')
plt.show()
elif self.dim == 2:
X, Y = np.meshgrid(x_grid, y_grid, indexing='ij')
XI = np.full_like(X, xi0)
ETA = np.full_like(Y, eta0)
symbol_vals = self.p_func(X, Y, XI, ETA)
plt.pcolormesh(X, Y, np.abs(symbol_vals), shading='auto')
plt.colorbar(label='|Symbol|')
plt.xlabel('x')
plt.ylabel('y')
plt.title(f'Symbol Amplitude at ξ={xi0}, η={eta0}')
plt.show()
def visualize_phase(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0):
"""
Plot the phase (argument) of the pseudodifferential operator's symbol p(x, ξ) or p(x, y, ξ, η).
This visualization helps in understanding the oscillatory behavior and regularity
properties of the operator in phase space. The phase is displayed modulo 2π using
a cyclic colormap ('twilight') to emphasize its periodic nature.
Parameters
----------
x_grid : ndarray
1D array of spatial coordinates (x).
xi_grid : ndarray
1D array of frequency coordinates (ξ).
y_grid : ndarray, optional
2D spatial grid for y-coordinate (in 2D problems). Default is None.
eta_grid : ndarray, optional
2D frequency grid for η (in 2D problems). Not used directly but kept for API consistency.
xi0 : float, optional
Fixed value of ξ for slicing in 2D visualization. Default is 0.0.
eta0 : float, optional
Fixed value of η for slicing in 2D visualization. Default is 0.0.
Notes:
- In 1D: Displays arg(p(x, ξ)) over the (x, ξ) phase plane.
- In 2D: Displays arg(p(x, y, ξ₀, η₀)) for fixed frequency values (ξ₀, η₀).
- Uses plt.pcolormesh with 'twilight' colormap to represent angles from -π to π.
Raises:
- NotImplementedError: If the spatial dimension is not 1D or 2D.
"""
if self.dim == 1:
X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij')
symbol_vals = self.p_func(X, XI)
plt.pcolormesh(X, XI, np.angle(symbol_vals), shading='auto', cmap='twilight')
plt.colorbar(label='arg(Symbol) [rad]')
plt.xlabel('x')
plt.ylabel('ξ')
plt.title('Phase Portrait (arg p(x, ξ))')
plt.show()
elif self.dim == 2:
X, Y = np.meshgrid(x_grid, y_grid, indexing='ij')
XI = np.full_like(X, xi0)
ETA = np.full_like(Y, eta0)
symbol_vals = self.p_func(X, Y, XI, ETA)
plt.pcolormesh(X, Y, np.angle(symbol_vals), shading='auto', cmap='twilight')
plt.colorbar(label='arg(Symbol) [rad]')
plt.xlabel('x')
plt.ylabel('y')
plt.title(f'Phase Portrait at ξ={xi0}, η={eta0}')
plt.show()
def visualize_characteristic_set(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0, levels=[1e-1]):
"""
Visualize the characteristic set of the pseudo-differential symbol, defined as the approximate zero set p(x, ξ) ≈ 0.
In microlocal analysis, the characteristic set is the locus of points in phase space (x, ξ) where the symbol p(x, ξ) vanishes,
playing a key role in understanding propagation of singularities and wavefronts.
Parameters
----------
x_grid : ndarray
Spatial grid values (1D array) for plotting in 1D or evaluation point in 2D.
xi_grid : ndarray
Frequency variable grid values (1D array) used to construct the frequency domain.
x0 : float, optional
Fixed spatial coordinate in 2D case for evaluating the symbol at a specific x position.
y0 : float, optional
Fixed spatial coordinate in 2D case for evaluating the symbol at a specific y position.
Notes
-----
- For 1D, this method plots the contour of |p(x, ξ)| = ε with ε = 1e-5 over the (x, ξ) plane.
- For 2D, it evaluates the symbol at fixed (x₀, y₀) and plots the characteristic set in the (ξ, η) frequency plane.
- This visualization helps identify directions of degeneracy or hypoellipticity of the operator.
Raises
------
NotImplementedError
If called on a solver with dimensionality other than 1D or 2D.
Displays
------
A matplotlib contour plot showing either:
- The characteristic curve in the (x, ξ) phase plane (1D),
- The characteristic surface slice in the (ξ, η) frequency plane at (x₀, y₀) (2D).
"""
if self.dim == 1:
x_grid = np.asarray(x_grid)
xi_grid = np.asarray(xi_grid)
X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij')
symbol_vals = self.p_func(X, XI)
plt.contour(X, XI, np.abs(symbol_vals), levels=levels, colors='red')
plt.xlabel('x')
plt.ylabel('ξ')
plt.title('Characteristic Set (p(x, ξ) ≈ 0)')
plt.grid(True)
plt.show()
elif self.dim == 2:
if eta_grid is None:
raise ValueError("eta_grid must be provided for 2D visualization.")
xi_grid = np.asarray(xi_grid)
eta_grid = np.asarray(eta_grid)
xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij')
symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2)
plt.contour(xi_grid, eta_grid, np.abs(symbol_vals), levels=levels, colors='red')
plt.xlabel('ξ')
plt.ylabel('η')
plt.title(f'Characteristic Set at x={x0}, y={y0}')
plt.grid(True)
plt.show()
else:
raise NotImplementedError("Only 1D/2D characteristic sets supported.")
def visualize_characteristic_gradient(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0):
"""
Visualize the norm of the gradient of the symbol in phase space.
This method computes the magnitude of the gradient |∇p| of a pseudo-differential
symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. The resulting colormap reveals
regions where the symbol varies rapidly or remains nearly stationary,
which is particularly useful for analyzing characteristic sets.
Parameters
----------
x_grid : numpy.ndarray
1D array of spatial coordinates for the x-direction.
xi_grid : numpy.ndarray
1D array of frequency coordinates (ξ).
y_grid : numpy.ndarray, optional
1D array of spatial coordinates for the y-direction (used in 2D mode). Default is None.
eta_grid : numpy.ndarray, optional
1D array of frequency coordinates (η) for the 2D case. Default is None.
x0 : float, optional
Fixed x-coordinate for evaluating the symbol in 2D. Default is 0.0.
y0 : float, optional
Fixed y-coordinate for evaluating the symbol in 2D. Default is 0.0.
Returns
-------
None
Displays a 2D colormap of |∇p| over the relevant phase-space domain.
Notes
-----
- In 1D, the full gradient ∇p = (∂ₓp, ∂ξp) is computed over the (x, ξ) grid.
- In 2D, the gradient ∇p = (∂ξp, ∂ηp) is computed at a fixed spatial point (x₀, y₀) over the (ξ, η) grid.
- Numerical differentiation is performed using `np.gradient`.
- High values of |∇p| indicate rapid variation of the symbol, while low values typically suggest characteristic regions.
"""
if self.dim == 1:
X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij')
symbol_vals = self.p_func(X, XI)
grad_x = np.gradient(symbol_vals, axis=0)
grad_xi = np.gradient(symbol_vals, axis=1)
grad_norm = np.sqrt(grad_x**2 + grad_xi**2)
plt.pcolormesh(X, XI, grad_norm, cmap='inferno', shading='auto')
plt.colorbar(label='|∇p|')
plt.title('Gradient Norm (High Near Zeros)')
plt.grid(True)
plt.show()
elif self.dim == 2:
xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij')
symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2)
grad_xi = np.gradient(symbol_vals, axis=0)
grad_eta = np.gradient(symbol_vals, axis=1)
grad_norm = np.sqrt(grad_xi**2 + grad_eta**2)
plt.pcolormesh(xi_grid, eta_grid, grad_norm, cmap='inferno', shading='auto')
plt.colorbar(label='|∇p|')
plt.title(f'Gradient Norm at x={x0}, y={y0}')
plt.grid(True)
plt.show()
def visualize_dynamic_wavefront(self, x_grid, t_grid, y_grid=None, xi0=5.0, eta0=0.0):
"""
Create an animation of a monochromatic wave evolving under the pseudo-differential operator.
This method visualizes the time evolution of a wavefront governed by the symbol of the
pseudo-differential operator. The wave is initialized as a monochromatic solution with
fixed spatial frequencies and evolves according to the dispersion relation
ω = p(x, ξ), where p is the symbol of the operator and (ξ, η) are spatial frequencies.
Parameters
----------
x_grid : array_like
1D array representing the spatial grid in the x-direction.
t_grid : array_like
1D array of time points used to generate the animation frames.
y_grid : array_like, optional
1D array representing the spatial grid in the y-direction (required for 2D operators).
xi0 : float, optional
Initial spatial frequency in the x-direction. Default is 5.0.
eta0 : float, optional
Initial spatial frequency in the y-direction (used in 2D). Default is 0.0.
Returns
-------
matplotlib.animation.FuncAnimation
Animation object showing the time evolution of the monochromatic wave.
Notes
-----
- In 1D, visualizes the wave u(x, t) = cos(ξ₀·x − ω·t).
- In 2D, visualizes u(x, y, t) = cos(ξ₀·x + η₀·y − ω·t).
- The frequency ω is computed as the symbol evaluated at the fixed frequency components.
- If the symbol depends on space, it is evaluated at the origin (x=0, y=0).
Raises
------
ValueError
If `y_grid` is not provided while the operator is 2D.
NotImplementedError
If the spatial dimension of the operator is neither 1 nor 2.
"""
if self.dim == 1:
# Calculer omega a partir du symbole
from sympy import symbols, N
x = self.vars_x[0]
xi = symbols('xi', real=True)
try:
omega_symbolic = self.expr.subs({x: 0, xi: xi0})
except:
omega_symbolic = self.expr.subs(xi, xi0)
omega_val = float(N(omega_symbolic.as_real_imag()[0]))
# Preparer les donnees pour l'animation
X, T = np.meshgrid(x_grid, t_grid, indexing='ij')
# Initialiser la figure et l'axe
fig, ax = plt.subplots()
ax.set_xlim(x_grid.min(), x_grid.max())
ax.set_ylim(-1.1, 1.1)
ax.set_xlabel('x')
ax.set_ylabel('u(x, t)')
ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t)')
ax.grid(True)
line, = ax.plot([], [], lw=2)
# Pre-calculer les temps
T_vals = t_grid
def animate(frame):
t = T_vals[frame] if frame < len(T_vals) else T_vals[-1]
U = np.cos(xi0 * x_grid - omega_val * t)
line.set_data(x_grid, U)
ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t) at t={t:.2f}')
return line,
ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=True, repeat=True)
plt.show()
return ani
elif self.dim == 2:
# Calculer omega a partir du symbole
from sympy import symbols, N
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
try:
omega_symbolic = self.expr.subs({x: 0, y: 0, xi: xi0, eta: eta0})
except:
omega_symbolic = self.expr.subs({xi: xi0, eta: eta0})
omega_val = float(N(omega_symbolic.as_real_imag()[0]))
if y_grid is None:
raise ValueError("y_grid must be provided for 2D visualization.")
# Creer les grilles
X, Y = np.meshgrid(x_grid, y_grid, indexing='ij')
# Initialiser la figure
fig, ax = plt.subplots()
ax.set_xlabel('x')
ax.set_ylabel('y')
# Premiere frame
t0 = t_grid[0] if len(t_grid) > 0 else 0.0
U = np.cos(xi0 * X + eta0 * Y - omega_val * t0)
extent = [x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()]
im = ax.imshow(U, extent=extent, aspect='equal', origin='lower', cmap='seismic', vmin=-1, vmax=1)
ax.set_title(f'Dynamic 2D Wavefront at t={t0:.2f}')
fig.colorbar(im, ax=ax, label='u(x, y, t)')
# Pre-calculer les temps
T_vals = t_grid
def animate(frame):
t = T_vals[frame] if frame < len(T_vals) else T_vals[-1]
U = np.cos(xi0 * X + eta0 * Y - omega_val * t)
im.set_array(U)
ax.set_title(f'Dynamic 2D Wavefront at t={t:.2f}')
return [im]
ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=False, repeat=True)
plt.show()
return ani
def simulate_evolution(self, x_grid, t_grid, y_grid=None,
initial_condition=None, initial_velocity=None,
solver_params=None, component='real'):
"""
Simulate and animate the time evolution of an initial condition under a pseudo-differential operator.
This method solves the PDE governed by the action of a pseudo-differential operator `p(x, D)`,
either as a first-order or second-order time evolution, and generates an animation of the resulting
wave propagation. The order of the PDE depends on whether an initial velocity is provided.
Parameters
----------
x_grid : numpy.ndarray
Spatial grid in the x-direction.
t_grid : numpy.ndarray
Temporal grid for the simulation.
y_grid : numpy.ndarray, optional
Spatial grid in the y-direction (used for 2D problems).
initial_condition : callable, optional
Function specifying the initial condition u₀(x) or u₀(x, y).
initial_velocity : callable, optional
Function specifying the initial velocity ∂ₜu₀(x) or ∂ₜu₀(x, y). If provided, a second-order
PDE is solved.
solver_params : dict, optional
Additional keyword arguments passed to the PDE solver.
component : {'real', 'imag', 'abs', 'angle'}
Specifies which component of the solution to animate.
Returns
-------
matplotlib.animation.FuncAnimation
Animation object showing the evolution of the solution over time.
Notes
-----
- Solves the PDE:
- First-order: ∂ₜu = -i ⋅ p(x, D) u
- Second-order: ∂²ₜu = -p(x, D)² u
- Supports 1D and 2D simulations depending on the presence of `y_grid`.
- The solution is animated using the specified component view.
- Useful for visualizing wave propagation and dispersive effects driven by pseudo-differential operators.
"""
if solver_params is None:
solver_params = {}
# --- 1. Symbolic variables ---
t = symbols('t', real=True)
u_sym = Function('u')
is_second_order = initial_velocity is not None
if self.dim == 1:
x, = self.vars_x
xi = symbols('xi', real=True)
u = u_sym(t, x)
if is_second_order:
eq = Eq(diff(u, t, 2), psiOp(self.expr, u))
else:
eq = Eq(diff(u, t), psiOp(self.expr, u))
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
u = u_sym(t, x, y)
if is_second_order:
eq = Eq(diff(u, t, 2), psiOp(self.expr, u))
else:
eq = Eq(diff(u, t), psiOp(self.expr, u))
else:
raise NotImplementedError("Only 1D and 2D are supported.")
# --- 2. Create the solver ---
solver = PDESolver(eq)
params = {
'Lx': x_grid.max() - x_grid.min(),
'Nx': len(x_grid),
'Lt': t_grid.max() - t_grid.min(),
'Nt': len(t_grid),
'boundary_condition': 'periodic',
'n_frames': min(100, len(t_grid))
}
if self.dim == 2:
params['Ly'] = y_grid.max() - y_grid.min()
params['Ny'] = len(y_grid)
params.update(solver_params)
# --- 3. Initial condition ---
if initial_condition is None:
raise ValueError("initial_condition is None. Please provide a function u₀(x) or u₀(x, y) as the initial condition.")
params['initial_condition'] = initial_condition
if is_second_order:
params['initial_velocity'] = initial_velocity
# --- 4. Solving ---
print("⚙️ Solving the evolution equation (order {} in time)...".format(2 if is_second_order else 1))
solver.setup(**params)
solver.solve()
print("✅ Solving completed.")
# --- 5. Animation ---
print("🎞️ Creating the animation...")
ani = solver.animate(component=component)
return ani
def plot_hamiltonian_flow(self, x0=0.0, xi0=5.0, y0=0.0, eta0=0.0, tmax=1.0, n_steps=100, show_field=True):
"""
Integrate and plot the Hamiltonian trajectories of the symbol in phase space.
This method numerically integrates the Hamiltonian vector field derived from
the operator's symbol to visualize how singularities propagate under the flow.
It supports both 1D and 2D problems.
Parameters
----------
x0, xi0 : float
Initial position and frequency (momentum) in 1D.
y0, eta0 : float, optional
Initial position and frequency in 2D; defaults to zero.
tmax : float
Final integration time for the ODE solver.
n_steps : int
Number of time steps used in the integration.
Notes
-----
- The Hamiltonian vector field is obtained from the symplectic flow of the symbol.
- If the field is complex-valued, only its real part is used for integration.
- In 1D, the trajectory is plotted in (x, ξ) phase space.
- In 2D, the spatial trajectory (x(t), y(t)) is shown along with instantaneous
momentum vectors (ξ(t), η(t)) using a quiver plot.
Raises
------
NotImplementedError
If the spatial dimension is not 1D or 2D.
Displays
--------
matplotlib plot
Phase space trajectory(ies) showing the evolution of position and momentum
under the Hamiltonian dynamics.
"""
def make_real(expr):
from sympy import re, simplify
expr = expr.doit(deep=True)
return simplify(re(expr))
H = self.symplectic_flow()
if any(im(H[k]) != 0 for k in H):
print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.")
if self.dim == 1:
x, = self.vars_x
xi = symbols('xi', real=True)
dxdt_expr = make_real(H['dx/dt'])
dxidt_expr = make_real(H['dxi/dt'])
dxdt = lambdify((x, xi), dxdt_expr, 'numpy')
dxidt = lambdify((x, xi), dxidt_expr, 'numpy')
def hamilton(t, Y):
x, xi = Y
return [dxdt(x, xi), dxidt(x, xi)]
sol = solve_ivp(hamilton, [0, tmax], [x0, xi0], t_eval=np.linspace(0, tmax, n_steps))
if sol.status != 0:
print(f"⚠️ Integration warning: {sol.message}")
n_points = sol.y.shape[1]
if n_points < n_steps:
print(f"⚠️ Only {n_points} frames computed. Adjusting animation.")
n_steps = n_points
x_vals, xi_vals = sol.y
plt.plot(x_vals, xi_vals)
plt.xlabel("x")
plt.ylabel("ξ")
plt.title("Hamiltonian Flow in Phase Space (1D)")
plt.grid(True)
plt.show()
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy')
dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy')
dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy')
detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy')
def hamilton(t, Y):
x, y, xi, eta = Y
return [
dxdt(x, y, xi, eta),
dydt(x, y, xi, eta),
dxidt(x, y, xi, eta),
detadt(x, y, xi, eta)
]
sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0], t_eval=np.linspace(0, tmax, n_steps))
if sol.status != 0:
print(f"⚠️ Integration warning: {sol.message}")
n_points = sol.y.shape[1]
if n_points < n_steps:
print(f"⚠️ Only {n_points} frames computed. Adjusting animation.")
n_steps = n_points
x_vals, y_vals, xi_vals, eta_vals = sol.y
plt.plot(x_vals, y_vals, label='Position')
plt.quiver(x_vals, y_vals, xi_vals, eta_vals, scale=20, width=0.003, alpha=0.5, color='r')
# Vector field of the flow (optional)
if show_field:
X, Y = np.meshgrid(np.linspace(min(x_vals), max(x_vals), 20),
np.linspace(min(y_vals), max(y_vals), 20))
XI, ETA = xi0 * np.ones_like(X), eta0 * np.ones_like(Y)
U = dxdt(X, Y, XI, ETA)
V = dydt(X, Y, XI, ETA)
plt.quiver(X, Y, U, V, color='gray', alpha=0.2, scale=30, width=0.002)
plt.xlabel("x")
plt.ylabel("y")
plt.title("Hamiltonian Flow in Phase Space (2D)")
plt.legend()
plt.grid(True)
plt.axis('equal')
plt.show()
def plot_symplectic_vector_field(self, xlim=(-2, 2), klim=(-5, 5), density=30):
"""
Visualize the symplectic vector field (Hamiltonian vector field) associated with the operator's symbol.
The plotted vector field corresponds to (∂_ξ p, -∂_x p), where p(x, ξ) is the principal symbol
of the pseudo-differential operator. This field governs the bicharacteristic flow in phase space.
Parameters
----------
xlim : tuple of float
Range for spatial variable x, as (x_min, x_max).
klim : tuple of float
Range for frequency variable ξ, as (ξ_min, ξ_max).
density : int
Number of grid points per axis for the visualization grid.
Raises
------
NotImplementedError
If called on a 2D operator (currently only 1D implementation available).
Notes
-----
- Only supports one-dimensional operators.
- Uses symbolic differentiation to compute ∂_ξ p and ∂_x p.
- Numerical evaluation is done via lambdify with NumPy backend.
- Visualization uses matplotlib quiver plot to show vector directions.
"""
x_vals = np.linspace(*xlim, density)
xi_vals = np.linspace(*klim, density)
X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij')
if self.dim != 1:
raise NotImplementedError("Only 1D version implemented.")
x, = self.vars_x
xi = symbols('xi', real=True)
H = self.symplectic_flow()
dxdt = lambdify((x, xi), simplify(H['dx/dt']), 'numpy')
dxidt = lambdify((x, xi), simplify(H['dxi/dt']), 'numpy')
U = dxdt(X, XI)
V = dxidt(X, XI)
plt.quiver(X, XI, U, V, scale=10, width=0.005)
plt.xlabel('x')
plt.ylabel(r'$\xi$')
plt.title("Symplectic Vector Field (1D)")
plt.grid(True)
plt.show()
def visualize_micro_support(self, xlim=(-2, 2), klim=(-10, 10), threshold=1e-3, density=300):
"""
Visualize the micro-support of the operator by plotting the inverse of the symbol magnitude 1 / |p(x, ξ)|.
The micro-support provides insight into the singularities of a pseudo-differential operator
in phase space (x, ξ). Regions where |p(x, ξ)| is small correspond to large values in 1/|p(x, ξ)|,
highlighting areas of significant operator influence or singularity.
Parameters
----------
xlim : tuple
Spatial domain limits (x_min, x_max).
klim : tuple
Frequency domain limits (ξ_min, ξ_max).
threshold : float
Threshold below which |p(x, ξ)| is considered effectively zero; used for numerical stability.
density : int
Number of grid points along each axis for visualization resolution.
Raises
------
NotImplementedError
If called on a solver with dimension greater than 1 (only 1D visualization is supported).
Notes
-----
- This method evaluates the symbol p(x, ξ) over a grid and plots its reciprocal to emphasize
regions where the symbol is near zero.
- A small constant (1e-10) is added to the denominator to avoid division by zero.
- The resulting plot helps identify characteristic sets and wavefront set approximations.
"""
if self.dim != 1:
raise NotImplementedError("Only 1D micro-support visualization implemented.")
x_vals = np.linspace(*xlim, density)
xi_vals = np.linspace(*klim, density)
X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij')
Z = np.abs(self.p_func(X, XI))
plt.contourf(X, XI, 1 / (Z + 1e-10), levels=100, cmap='inferno')
plt.colorbar(label=r'$1/|p(x,\xi)|$')
plt.xlabel('x')
plt.ylabel(r'$\xi$')
plt.title("Micro-Support Estimate (1/|Symbol|)")
plt.show()
def group_velocity_field(self, xlim=(-2, 2), klim=(-10, 10), density=30):
"""
Plot the group velocity field ∇_ξ p(x, ξ) for 1D pseudo-differential operators.
The group velocity represents the speed at which waves of different frequencies propagate
in a dispersive medium. It is defined as the gradient of the symbol p(x, ξ) with respect
to the frequency variable ξ.
Parameters
----------
xlim : tuple of float
Spatial domain limits (x-axis).
klim : tuple of float
Frequency domain limits (ξ-axis).
density : int
Number of grid points per axis used for visualization.
Raises
------
NotImplementedError
If called on a 2D operator, since this visualization is only implemented for 1D.
Notes
-----
- This method visualizes the vector field (∂p/∂ξ) in phase space.
- Used for analyzing wave propagation properties and dispersion relations.
- Requires symbolic expression self.expr depending on x and ξ.
"""
if self.dim != 1:
raise NotImplementedError("Only 1D group velocity visualization implemented.")
x, = self.vars_x
xi = symbols('xi', real=True)
dp_dxi = diff(self.expr, xi)
grad_func = lambdify((x, xi), dp_dxi, 'numpy')
x_vals = np.linspace(*xlim, density)
xi_vals = np.linspace(*klim, density)
X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij')
V = grad_func(X, XI)
plt.quiver(X, XI, np.ones_like(V), V, scale=10, width=0.004)
plt.xlabel('x')
plt.ylabel(r'$\xi$')
plt.title("Group Velocity Field (1D)")
plt.grid(True)
plt.show()
def animate_singularity(self, xi0=5.0, eta0=0.0, x0=0.0, y0=0.0,
tmax=4.0, n_frames=100, projection=None):
"""
Animate the propagation of a singularity under the Hamiltonian flow.
This method visualizes how a singularity (x₀, y₀, ξ₀, η₀) evolves in phase space
according to the Hamiltonian dynamics induced by the principal symbol of the operator.
The animation integrates the Hamiltonian equations of motion and supports various projections:
position (x-y), frequency (ξ-η), or mixed phase space coordinates.
Parameters
----------
xi0, eta0 : float
Initial frequency components (ξ₀, η₀).
x0, y0 : float
Initial spatial coordinates (x₀, y₀).
tmax : float
Total time of integration (final animation time).
n_frames : int
Number of frames in the resulting animation.
projection : str or None
Type of projection to display:
- 'position' : x vs y (or x alone in 1D)
- 'frequency': ξ vs η (or ξ alone in 1D)
- 'phase' : mixed coordinates like x vs ξ or x vs η
If None, defaults to 'phase' in 1D and 'position' in 2D.
Returns
-------
matplotlib.animation.FuncAnimation
Animation object that can be displayed interactively in Jupyter notebooks or saved as a video.
Notes
-----
- In 1D, only one spatial and one frequency variable are used.
- Complex-valued Hamiltonian fields are truncated to their real parts for integration.
- Trajectories are shown with both instantaneous position (dot) and full path (dashed line).
"""
rc('animation', html='jshtml')
def make_real(expr):
from sympy import re, simplify
expr = expr.doit(deep=True)
return simplify(re(expr))
H = self.symplectic_flow()
H = {k: v.doit(deep=True) for k, v in H.items()}
print("H = ", H)
if any(im(H[k]) != 0 for k in H):
print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.")
if self.dim == 1:
x, = self.vars_x
xi = symbols('xi', real=True)
dxdt = lambdify((x, xi), make_real(H['dx/dt']), 'numpy')
dxidt = lambdify((x, xi), make_real(H['dxi/dt']), 'numpy')
def hamilton(t, Y):
x, xi = Y
return [dxdt(x, xi), dxidt(x, xi)]
sol = solve_ivp(hamilton, [0, tmax], [x0, xi0],
t_eval=np.linspace(0, tmax, n_frames))
if sol.status != 0:
print(f"⚠️ Integration warning: {sol.message}")
n_points = sol.y.shape[1]
if n_points < n_frames:
print(f"⚠️ Only {n_points} frames computed. Adjusting animation.")
n_frames = n_points
x_vals, xi_vals = sol.y
if projection is None:
projection = 'phase'
fig, ax = plt.subplots()
point, = ax.plot([], [], 'ro')
traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5)
if projection == 'phase':
ax.set_xlabel('x')
ax.set_ylabel(r'$\xi$')
ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1)
ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1)
def update(i):
point.set_data([x_vals[i]], [xi_vals[i]])
traj.set_data(x_vals[:i+1], xi_vals[:i+1])
return point, traj
elif projection == 'position':
ax.set_xlabel('x')
ax.set_ylabel('x')
ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1)
ax.set_ylim(np.min(x_vals) - 1, np.max(x_vals) + 1)
def update(i):
point.set_data([x_vals[i]], [x_vals[i]])
traj.set_data(x_vals[:i+1], x_vals[:i+1])
return point, traj
elif projection == 'frequency':
ax.set_xlabel(r'$\xi$')
ax.set_ylabel(r'$\xi$')
ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1)
ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1)
def update(i):
point.set_data([xi_vals[i]], [xi_vals[i]])
traj.set_data(xi_vals[:i+1], xi_vals[:i+1])
return point, traj
else:
raise ValueError("Invalid projection mode")
ax.set_title(f"1D Singularity Flow ({projection})")
ax.grid(True)
ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50)
plt.close(fig)
return ani
elif self.dim == 2:
x, y = self.vars_x
xi, eta = symbols('xi eta', real=True)
dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy')
dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy')
dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy')
detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy')
def hamilton(t, Y):
x, y, xi, eta = Y
return [
dxdt(x, y, xi, eta),
dydt(x, y, xi, eta),
dxidt(x, y, xi, eta),
detadt(x, y, xi, eta)
]
sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0],
t_eval=np.linspace(0, tmax, n_frames))
if sol.status != 0:
print(f"⚠️ Integration warning: {sol.message}")
n_points = sol.y.shape[1]
if n_points < n_frames:
print(f"⚠️ Only {n_points} frames computed. Adjusting animation.")
n_frames = n_points
x_vals, y_vals, xi_vals, eta_vals = sol.y
if projection is None:
projection = 'position'
fig, ax = plt.subplots()
point, = ax.plot([], [], 'ro')
traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5)
if projection == 'position':
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1)
ax.set_ylim(np.min(y_vals) - 1, np.max(y_vals) + 1)
def update(i):
point.set_data([x_vals[i]], [y_vals[i]])
traj.set_data(x_vals[:i+1], y_vals[:i+1])
return point, traj
elif projection == 'frequency':
ax.set_xlabel(r'$\xi$')
ax.set_ylabel(r'$\eta$')
ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1)
ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1)
def update(i):
point.set_data([xi_vals[i]], [eta_vals[i]])
traj.set_data(xi_vals[:i+1], eta_vals[:i+1])
return point, traj
elif projection == 'phase':
ax.set_xlabel('x')
ax.set_ylabel(r'$\eta$')
ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1)
ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1)
def update(i):
point.set_data([x_vals[i]], [eta_vals[i]])
traj.set_data(x_vals[:i+1], eta_vals[:i+1])
return point, traj
else:
raise ValueError("Invalid projection mode")
ax.set_title(f"2D Singularity Flow ({projection})")
ax.grid(True)
ax.axis('equal')
ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50)
plt.close(fig)
return ani
def interactive_symbol_analysis(pseudo_op,
xlim=(-2, 2), ylim=(-2, 2),
xi_range=(0.1, 5), eta_range=(-5, 5),
density=100):
"""
Launch an interactive dashboard for symbol exploration using ipywidgets.
This function provides a user-friendly interface to visualize various aspects of the pseudo-differential operator's symbol.
It supports multiple visualization modes in both 1D and 2D, including group velocity fields, micro-support estimates,
symplectic vector fields, symbol amplitude/phase, cotangent fiber structure, characteristic sets, wavefront sets,
and Hamiltonian flows.
Parameters
----------
pseudo_op : PseudoDifferentialOperator
The pseudo-differential operator whose symbol is to be analyzed interactively.
xlim, ylim : tuple of float
Spatial domain limits along x and y axes respectively.
xi_range, eta_range : tuple
Frequency domain limits along ξ and η axes respectively.
density : int
Number of points per axis used to construct the evaluation grid. Controls resolution.
Notes
-----
- In 1D mode, sliders control the fixed frequency (ξ₀) and spatial position (x₀).
- In 2D mode, additional sliders control the second frequency component (η₀) and second spatial coordinate (y₀).
- Visualization updates dynamically as parameters are adjusted via sliders or dropdown menus.
- Supported visualization modes:
'Group Velocity Field' : ∇_ξ p(x,ξ) or ∇_{ξ,η} p(x,y,ξ,η)
'Micro-Support (1/|p|)' : Reciprocal of symbol magnitude
'Symplectic Vector Field' : (∇_ξ p, -∇_x p) or similar in 2D
'Symbol Amplitude' : |p(x,ξ)| or |p(x,y,ξ,η)|
'Symbol Phase' : arg(p(x,ξ)) or similar in 2D
'Cotangent Fiber' : Structure of symbol over frequency space at fixed x
'Characteristic Set' : Zero set approximation {p ≈ 0}
'Characteristic Gradient' : |∇p(x, ξ)| or |∇p(x₀, y₀, ξ, η)|
'Wavefront Set' : High-frequency singularities detected via symbol interaction
'Hamiltonian Flow' : Trajectories generated by the Hamiltonian vector field
Raises
------
NotImplementedError
If the spatial dimension is not 1D or 2D.
Prints
------
Interactive matplotlib figures with dynamic updates based on widget inputs.
"""
dim = pseudo_op.dim
expr = pseudo_op.expr
vars_x = pseudo_op.vars_x
mode_selector = Dropdown(
options=[
'Group Velocity Field',
'Micro-Support (1/|p|)',
'Symplectic Vector Field',
'Symbol Amplitude',
'Symbol Phase',
'Cotangent Fiber',
'Characteristic Set',
'Characteristic Gradient',
'Wavefront Set',
'Hamiltonian Flow',
],
value='Group Velocity Field',
description='Mode:'
)
x_vals = np.linspace(*xlim, density)
if dim == 2:
y_vals = np.linspace(*ylim, density)
if dim == 1:
x, = vars_x
xi = symbols('xi', real=True)
grad_func = lambdify((x, xi), diff(expr, xi), 'numpy')
symplectic_func = lambdify((x, xi), [diff(expr, xi), -diff(expr, x)], 'numpy')
symbol_func = lambdify((x, xi), expr, 'numpy')
def plot_1d(mode, xi0, x0):
X = x_vals[:, None]
if mode == 'Group Velocity Field':
V = grad_func(X, xi0)
plt.quiver(X, V, np.ones_like(V), V, scale=10, width=0.004)
plt.title(f'Group Velocity Field at ξ={xi0:.2f}')
elif mode == 'Micro-Support (1/|p|)':
Z = 1 / (np.abs(symbol_func(X, xi0)) + 1e-10)
plt.plot(x_vals, Z)
plt.title(f'Micro-Support (1/|p|) at ξ={xi0:.2f}')
elif mode == 'Symplectic Vector Field':
U, V = symplectic_func(X, xi0)
plt.quiver(X, V, U, V, scale=10, width=0.004)
plt.title(f'Symplectic Field at ξ={xi0:.2f}')
elif mode == 'Symbol Amplitude':
Z = np.abs(symbol_func(X, xi0))
plt.plot(x_vals, Z)
plt.title(f'Symbol Amplitude |p(x,ξ)| at ξ={xi0:.2f}')
elif mode == 'Symbol Phase':
Z = np.angle(symbol_func(X, xi0))
plt.plot(x_vals, Z)
plt.title(f'Symbol Phase arg(p(x,ξ)) at ξ={xi0:.2f}')
elif mode == 'Cotangent Fiber':
pseudo_op.visualize_fiber(x_vals, np.linspace(*xi_range, density), x0=x0)
elif mode == 'Characteristic Set':
pseudo_op.visualize_characteristic_set(x_vals, np.linspace(*xi_range, density), x0=x0)
elif mode == 'Characteristic Gradient':
pseudo_op.visualize_characteristic_gradient(x_vals, np.linspace(*xi_range, density), x0=x0)
elif mode == 'Wavefront Set':
pseudo_op.visualize_wavefront(x_vals, np.linspace(*xi_range, density), xi0=xi0)
elif mode == 'Hamiltonian Flow':
pseudo_op.plot_hamiltonian_flow(x0=x0, xi0=xi0)
interact(plot_1d,
mode=mode_selector,
xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'),
x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀'))
elif dim == 2:
x, y = vars_x
xi, eta = symbols('xi eta', real=True)
grad_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy')
symplectic_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy')
symbol_func = lambdify((x, y, xi, eta), expr, 'numpy')
def plot_2d(mode, xi0, eta0, x0, y0):
X, Y = np.meshgrid(x_vals, y_vals, indexing='ij')
if mode == 'Group Velocity Field':
U, V = grad_func(X, Y, xi0, eta0)
plt.quiver(X, Y, U, V, scale=10, width=0.004)
plt.title(f'Group Velocity Field at ξ={xi0:.2f}, η={eta0:.2f}')
elif mode == 'Micro-Support (1/|p|)':
Z = 1 / (np.abs(symbol_func(X, Y, xi0, eta0)) + 1e-10)
plt.pcolormesh(X, Y, Z, shading='auto', cmap='inferno')
plt.colorbar(label='1/|p|')
plt.title(f'Micro-Support at ξ={xi0:.2f}, η={eta0:.2f}')
elif mode == 'Symplectic Vector Field':
U, V = symplectic_func(X, Y, xi0, eta0)
plt.quiver(X, Y, U, V, scale=10, width=0.004)
plt.title(f'Symplectic Field at ξ={xi0:.2f}, η={eta0:.2f}')
elif mode == 'Symbol Amplitude':
Z = np.abs(symbol_func(X, Y, xi0, eta0))
plt.pcolormesh(X, Y, Z, shading='auto')
plt.colorbar(label='|p(x,y,ξ,η)|')
plt.title(f'Symbol Amplitude at ξ={xi0:.2f}, η={eta0:.2f}')
elif mode == 'Symbol Phase':
Z = np.angle(symbol_func(X, Y, xi0, eta0))
plt.pcolormesh(X, Y, Z, shading='auto', cmap='twilight')
plt.colorbar(label='arg(p)')
plt.title(f'Symbol Phase at ξ={xi0:.2f}, η={eta0:.2f}')
elif mode == 'Cotangent Fiber':
pseudo_op.visualize_fiber(np.linspace(*xi_range, density), np.linspace(*eta_range, density),
x0=x0, y0=y0)
elif mode == 'Characteristic Set':
pseudo_op.visualize_characteristic_set(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density),
y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0)
elif mode == 'Characteristic Gradient':
pseudo_op.visualize_characteristic_gradient(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density),
y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0)
elif mode == 'Wavefront Set':
pseudo_op.visualize_wavefront(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density),
y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), xi0=xi0, eta0=eta0)
elif mode == 'Hamiltonian Flow':
pseudo_op.plot_hamiltonian_flow(x0=x0, y0=y0, xi0=xi0, eta0=eta0)
interact(plot_2d,
mode=mode_selector,
xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'),
eta0=FloatSlider(min=eta_range[0], max=eta_range[1], step=0.1, value=1.0, description='η₀'),
x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀'),
y0=FloatSlider(min=ylim[0], max=ylim[1], step=0.1, value=0.0, description='y₀'))
class PDESolver:
"""
A partial differential equation (PDE) solver based on **spectral methods** using Fourier transforms.
This solver supports symbolic specification of PDEs via SymPy and numerical solution using high-order spectral techniques.
It is designed for both **linear and nonlinear time-dependent PDEs**, as well as **stationary pseudo-differential problems**.
Key Features:
-------------
- Symbolic PDE parsing using SymPy expressions
- 1D and 2D spatial domains with periodic boundary conditions
- Fourier-based spectral discretization with dealiasing
- Temporal integration schemes:
- Default exponential time stepping
- ETD-RK4 (Exponential Time Differencing Runge-Kutta of 4th order)
- Nonlinear terms handled through pseudo-spectral evaluation
- Built-in tools for:
- Visualization of solutions and error surfaces
- Symbol analysis of linear and pseudo-differential operators
- Microlocal analysis (e.g., wavefront set estimation, Hamiltonian flows)
- CFL condition checking and numerical stability diagnostics
Supported Operators:
--------------------
- Linear differential and pseudo-differential operators
- Nonlinear terms up to second order in derivatives
- Symbolic operator composition and adjoints
- Asymptotic inversion of elliptic operators for stationary problems
Example Usage:
--------------
>>> from PDESolver import *
>>> u = Function('u')
>>> t, x = symbols('t x')
>>> eq = Eq(diff(u(t, x), t), diff(u(t, x), x, 2) + u(t, x)**2)
>>> def initial(x): return np.sin(x)
>>> solver = PDESolver(eq)
>>> solver.setup(Lx=2*np.pi, Nx=128, Lt=1.0, Nt=1000, initial_condition=initial)
>>> solver.solve()
>>> ani = solver.animate()
>>> HTML(ani.to_jshtml()) # Display animation in Jupyter notebook
"""
def __init__(self, equation, time_scheme='default', dealiasing_ratio=2/3):
"""
Initialize the PDE solver with a given equation.
This method analyzes the input partial differential equation (PDE),
identifies the unknown function and its dependencies, determines whether
the problem is stationary or time-dependent, and prepares symbolic and
numerical structures for solving in spectral space.
Supported features:
- 1D and 2D problems
- Time-dependent and stationary equations
- Linear and nonlinear terms
- Pseudo-differential operators via `psiOp`
- Source terms and boundary conditions
The equation is parsed to extract linear, nonlinear, source, and
pseudo-differential components. Symbolic manipulation is used to derive
the Fourier representation of linear operators when applicable.
Parameters
----------
equation : sympy.Eq
The PDE expressed as a SymPy equation.
time_scheme : str
Temporal integration scheme:
- 'default' for exponential
- time-stepping or 'ETD-RK4' for fourth-order exponential
- time differencing Runge–Kutta.
dealiasing_ratio : float
Fraction of high-frequency modes to zero out
during dealiasing (e.g., 2/3 for standard truncation).
Attributes initialized:
- self.u: the unknown function (e.g., u(t, x))
- self.dim: spatial dimension (1 or 2)
- self.spatial_vars: list of spatial variables (e.g., [x] or [x, y])
- self.is_stationary: boolean indicating if the problem is stationary
- self.linear_terms: dictionary mapping derivative orders to coefficients
- self.nonlinear_terms: list of nonlinear expressions
- self.source_terms: list of source functions
- self.pseudo_terms: list of pseudo-differential operator expressions
- self.has_psi: boolean indicating presence of pseudo-differential operators
- self.fft / self.ifft: appropriate FFT routines based on spatial dimension
- self.kx, self.ky: symbolic wavenumber variables for Fourier space
Raises:
ValueError: If the equation does not contain exactly one unknown function,
if unsupported dimensions are detected, or invalid dependencies.
"""
self.time_scheme = time_scheme # 'default' or 'ETD-RK4'
self.dealiasing_ratio = dealiasing_ratio
print("\n*********************************")
print("* Partial differential equation *")
print("*********************************\n")
pprint(equation)
# Extract symbols and function from the equation
functions = equation.atoms(Function)
# Ignore the wrappers psiOp and Op
excluded_wrappers = {'psiOp', 'Op'}
# Extract the candidate fonctions (excluding wrappers)
candidate_functions = [
f for f in functions
if f.func.__name__ not in excluded_wrappers
]
# Keep only user functions (u(x), u(x, t), etc.)
candidate_functions = [
f for f in functions
if isinstance(f, AppliedUndef)
]
# Stationary detection: no dependence on t
self.is_stationary = all(
not any(str(arg) == 't' for arg in f.args)
for f in candidate_functions
)
if len(candidate_functions) != 1:
print("candidate_functions :", candidate_functions)
raise ValueError("The equation must contain exactly one unknown function")
self.u = candidate_functions[0]
self.u_eq = self.u
args = self.u.args
if self.is_stationary:
if len(args) not in (1, 2):
raise ValueError("Stationary problems must depend on 1 or 2 spatial variables")
self.spatial_vars = args
else:
if len(args) < 2 or len(args) > 3:
raise ValueError("The function must depend on t and at least one spatial variable (x [, y])")
self.t = args[0]
self.spatial_vars = args[1:]
self.dim = len(self.spatial_vars)
if self.dim == 1:
self.x = self.spatial_vars[0]
self.y = None
elif self.dim == 2:
self.x, self.y = self.spatial_vars
else:
raise ValueError("Only 1D and 2D problems are supported.")
if self.dim == 1:
self.fft = partial(fft, workers=FFT_WORKERS)
self.ifft = partial(ifft, workers=FFT_WORKERS)
else:
self.fft = partial(fft2, workers=FFT_WORKERS)
self.ifft = partial(ifft2, workers=FFT_WORKERS)
# Parse the equation
self.linear_terms = {}
self.nonlinear_terms = []
self.symbol_terms = []
self.source_terms = []
self.pseudo_terms = []
self.temporal_order = 0 # Order of the temporal derivative
self.linear_terms, self.nonlinear_terms, self.symbol_terms, self.source_terms, self.pseudo_terms = self.parse_equation(equation)
# flag : pseudo‑differential operator present ?
self.has_psi = bool(self.pseudo_terms)
if self.has_psi:
print('⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.')
self.is_spatial = False
for coeff, expr in self.pseudo_terms:
if expr.has(self.x) or (self.dim == 2 and expr.has(self.y)):
self.is_spatial = True
break
if self.dim == 1:
self.kx = symbols('kx')
elif self.dim == 2:
self.kx, self.ky = symbols('kx ky')
# Compute linear operator
if not self.is_stationary:
self.compute_linear_operator()
else:
self.psi_ops = []
for coeff, sym_expr in self.pseudo_terms:
psi = PseudoDifferentialOperator(sym_expr, self.spatial_vars, self.u, mode='symbol')
self.psi_ops.append((coeff, psi))
def parse_equation(self, equation):
"""
Parse the PDE to separate linear and nonlinear terms, symbolic operators (Op),
source terms, and pseudo-differential operators (psiOp).
This method rewrites the input equation in standard form (lhs - rhs = 0),
expands it, and classifies each term into one of the following categories:
- Linear terms involving derivatives or the unknown function u
- Nonlinear terms (products with u, powers of u, etc.)
- Symbolic pseudo-differential operators (Op)
- Source terms (independent of u)
- Pseudo-differential operators (psiOp)
Parameters
equation (sympy.Eq): The partial differential equation to be analyzed.
Can be provided as an Eq object or a sympy expression.
Returns:
tuple: A 5-tuple containing:
- linear_terms (dict): Mapping from derivative/function to coefficient.
- nonlinear_terms (list): List of terms classified as nonlinear.
- symbol_terms (list): List of (coefficient, symbolic operator) pairs.
- source_terms (list): List of terms independent of the unknown function.
- pseudo_terms (list): List of (coefficient, pseudo-differential symbol) pairs.
Notes:
- If `psiOp` is present in the equation, expansion is skipped for safety.
- When `psiOp` is used, only nonlinear terms, source terms, and possibly
a time derivative are allowed; other linear terms and symbolic operators
(Op) are forbidden.
- Classification logic includes:
- Detection of nonlinear structures like products or powers of u
- Mixed terms involving both u and its derivatives
- External symbolic operators (Op) and pseudo-differential operators (psiOp)
"""
def is_nonlinear_term(term, u_func):
# If the term contains functions (Abs, sin, exp, ...) applied to u
if term.has(u_func):
for sub in preorder_traversal(term):
if isinstance(sub, Function) and sub.has(u_func) and sub.func != u_func.func:
return True
# If the term contains a nonlinear power of u
if term.has(Pow):
for pow_term in term.atoms(Pow):
if pow_term.base == u_func and pow_term.exp != 1:
return True
# If the term is a product containing u and its derivative
if term.func == Mul:
factors = term.args
has_u = any((f.has(u_func) and not isinstance(f, Derivative) for f in factors))
has_derivative = any((isinstance(f, Derivative) and f.expr.func == u_func.func for f in factors))
if has_u and has_derivative:
return True
return False
print("\n********************")
print("* Equation parsing *")
print("********************\n")
if isinstance(equation, Eq):
lhs = equation.lhs - equation.rhs
else:
lhs = equation
print(f"\nEquation rewritten in standard form: {lhs}")
if lhs.has(psiOp):
print("⚠️ psiOp detected: skipping expansion for safety")
lhs_expanded = lhs
else:
lhs_expanded = expand(lhs)
print(f"\nExpanded equation: {lhs_expanded}")
linear_terms = {}
nonlinear_terms = []
symbol_terms = []
source_terms = []
pseudo_terms = []
for term in lhs_expanded.as_ordered_terms():
print(f"Analyzing term: {term}")
if isinstance(term, psiOp):
expr = term.args[0]
pseudo_terms.append((1, expr))
print(" --> Classified as pseudo linear term (psiOp)")
continue
# Otherwise, look for psiOp inside (general case)
if term.has(psiOp):
psiops = term.atoms(psiOp)
for psi in psiops:
try:
coeff = simplify(term / psi)
expr = psi.args[0]
pseudo_terms.append((coeff, expr))
print(" --> Classified as pseudo linear term (psiOp)")
except Exception as e:
print(f" ⚠️ Failed to extract psiOp coefficient in term: {term}")
print(f" Reason: {e}")
nonlinear_terms.append(term)
print(" --> Fallback: classified as nonlinear")
continue
if term.has(Op):
ops = term.atoms(Op)
for op in ops:
coeff = term / op
expr = op.args[0]
symbol_terms.append((coeff, expr))
print(" --> Classified as symbolic linear term (Op)")
continue
if is_nonlinear_term(term, self.u):
nonlinear_terms.append(term)
print(" --> Classified as nonlinear")
continue
derivs = term.atoms(Derivative)
if derivs:
deriv = derivs.pop()
coeff = term / deriv
linear_terms[deriv] = linear_terms.get(deriv, 0) + coeff
print(f" Derivative found: {deriv}")
print(" --> Classified as linear")
elif self.u in term.atoms(Function):
coeff = term.as_coefficients_dict().get(self.u, 1)
linear_terms[self.u] = linear_terms.get(self.u, 0) + coeff
print(" --> Classified as linear")
else:
source_terms.append(term)
print(" --> Classified as source term")
print(f"Final linear terms: {linear_terms}")
print(f"Final nonlinear terms: {nonlinear_terms}")
print(f"Symbol terms: {symbol_terms}")
print(f"Pseudo terms: {pseudo_terms}")
print(f"Source terms: {source_terms}")
if pseudo_terms:
# Check if a time derivative is present among the linear terms
has_time_derivative = any(
isinstance(term, Derivative) and self.t in [v for v, _ in term.variable_count]
for term in linear_terms
)
# Extract non-temporal linear terms
invalid_linear_terms = {
term: coeff for term, coeff in linear_terms.items()
if not (
isinstance(term, Derivative)
and self.t in [v for v, _ in term.variable_count]
)
and term != self.u # exclusion of the simple u term (without derivative)
}
if invalid_linear_terms or symbol_terms:
raise ValueError(
"When psiOp is used, only nonlinear terms, source terms, "
"and possibly a time derivative are allowed. "
"Other linear terms and Ops are forbidden."
)
return linear_terms, nonlinear_terms, symbol_terms, source_terms, pseudo_terms
def compute_linear_operator(self):
"""
Compute the symbolic Fourier representation L(k) of the linear operator
derived from the linear part of the PDE.
This method constructs a dispersion relation by applying each symbolic derivative
to a plane wave exp(i(k·x - ωt)) and extracting the resulting expression.
It handles arbitrary derivative combinations and includes symbolic and
pseudo-differential terms.
Steps:
-------
1. Construct a plane wave φ(x, t) = exp(i(k·x - ωt)).
2. Apply each term from self.linear_terms to φ.
3. Normalize by φ and simplify to obtain L(k).
4. Include symbolic terms (e.g., psiOp) if present.
5. Detect the temporal order from the dispersion relation.
6. Build the numerical function L(k) via lambdify.
Sets:
-----
- self.L_symbolic : sympy.Expr
Symbolic form of L(k).
- self.L : callable
Numerical function of L(kx[, ky]).
- self.omega : callable or None
Frequency root ω(k), if available.
- self.temporal_order : int
Order of time derivatives detected.
- self.psi_ops : list of (coeff, PseudoDifferentialOperator)
Pseudo-differential terms present in the equation.
Raises:
-------
ValueError if the dimension is unsupported or the dispersion relation fails.
"""
print("\n*******************************")
print("* Linear operator computation *")
print("*******************************\n")
# --- Step 1: symbolic variables ---
omega = symbols("omega")
if self.dim == 1:
kvars = [symbols("kx")]
space_vars = [self.x]
elif self.dim == 2:
kvars = symbols("kx ky")
space_vars = [self.x, self.y]
else:
raise ValueError("Only 1D and 2D are supported.")
kdict = dict(zip(space_vars, kvars))
self.k_symbols = kvars
# Plane wave expression
phase = sum(k * x for k, x in zip(kvars, space_vars)) - omega * self.t
plane_wave = exp(I * phase)
# --- Step 2: build lhs expression from linear terms ---
lhs = 0
for deriv, coeff in self.linear_terms.items():
if isinstance(deriv, Derivative):
total_factor = 1
for var, n in deriv.variable_count:
if var == self.t:
total_factor *= (-I * omega)**n
elif var in kdict:
total_factor *= (I * kdict[var])**n
else:
raise ValueError(f"Unknown variable {var} in derivative")
lhs += coeff * total_factor * plane_wave
elif deriv == self.u:
lhs += coeff * plane_wave
else:
raise ValueError(f"Unsupported linear term: {deriv}")
# --- Step 3: dispersion relation ---
equation = simplify(lhs / plane_wave)
print("\nCharacteristic equation before symbol treatment:")
pprint(equation)
print("\n--- Symbolic symbol analysis ---")
symb_omega = 0
symb_k = 0
for coeff, symbol in self.symbol_terms:
if symbol.has(omega):
# Ajouter directement les termes dépendant de omega
symb_omega += coeff * symbol
elif any(symbol.has(k) for k in self.k_symbols):
symb_k += coeff * symbol.subs(dict(zip(symbol.free_symbols, self.k_symbols)))
print(f"symb_omega: {symb_omega}")
print(f"symb_k: {symb_k}")
equation = equation + symb_omega + symb_k
print("\nRaw characteristic equation:")
pprint(equation)
# Temporal derivative order detection
try:
poly_eq = Eq(equation, 0)
poly = poly_eq.lhs.as_poly(omega)
self.temporal_order = poly.degree() if poly else 0
except Exception as e:
warnings.warn(f"Could not determine temporal order: {e}", RuntimeWarning)
self.temporal_order = 0
print(f"Temporal order from dispersion relation: {self.temporal_order}")
print('self.pseudo_terms = ', self.pseudo_terms)
if self.pseudo_terms:
coeff_time = 1
for term, coeff in self.linear_terms.items():
if isinstance(term, Derivative) and any(var == self.t for var, _ in term.variable_count):
coeff_time = coeff
print(f"✅ Time derivative coefficient detected: {coeff_time}")
self.psi_ops = []
for coeff, sym_expr in self.pseudo_terms:
# expr est le Sympy expr. différentiel, var_x la liste [x] ou [x,y]
psi = PseudoDifferentialOperator(sym_expr / coeff_time, self.spatial_vars, self.u, mode='symbol')
self.psi_ops.append((coeff, psi))
else:
dispersion = solve(Eq(equation, 0), omega)
if not dispersion:
raise ValueError("No solution found for omega")
print("\n--- Solutions found ---")
pprint(dispersion)
if self.temporal_order == 2:
omega_expr = simplify(sqrt(dispersion[0]**2))
self.omega_symbolic = omega_expr
self.omega = lambdify(self.k_symbols, omega_expr, "numpy")
self.L_symbolic = -omega_expr**2
else:
self.L_symbolic = -I * dispersion[0]
self.L = lambdify(self.k_symbols, self.L_symbolic, "numpy")
print("\n--- Final linear operator ---")
pprint(self.L_symbolic)
def linear_rhs(self, u, is_v=False):
"""
Apply the linear operator (in Fourier space) to the field u or v.
Parameters
----------
u : np.ndarray
Input solution array.
is_v : bool
Whether to apply the operator to v instead of u.
Returns
-------
np.ndarray
Result of applying the linear operator.
"""
if self.dim == 1:
self.symbol_u = np.array(self.L(self.KX), dtype=np.complex128)
self.symbol_v = self.symbol_u # même opérateur pour u et v
elif self.dim == 2:
self.symbol_u = np.array(self.L(self.KX, self.KY), dtype=np.complex128)
self.symbol_v = self.symbol_u
u_hat = self.fft(u)
u_hat *= self.symbol_v if is_v else self.symbol_u
u_hat *= self.dealiasing_mask
return self.ifft(u_hat)
def setup(self, Lx, Ly=None, Nx=None, Ny=None, Lt=1.0, Nt=100, boundary_condition='periodic',
initial_condition=None, initial_velocity=None, n_frames=100):
"""
Configure the spatial/temporal grid and initialize the solution field.
This method sets up the computational domain, initializes spatial and temporal grids,
applies boundary conditions, and prepares symbolic and numerical operators.
It also performs essential analyses such as:
- CFL condition verification (for stability)
- Symbol analysis (e.g., dispersion relation, regularity)
- Wave propagation analysis for second-order equations
If pseudo-differential operators (ψOp) are present, symbolic analysis is skipped
in favor of interactive exploration via `interactive_symbol_analysis`.
Parameters
----------
Lx : float
Size of the spatial domain along x-axis.
Ly : float, optional
Size of the spatial domain along y-axis (for 2D problems).
Nx : int
Number of spatial points along x-axis.
Ny : int, optional
Number of spatial points along y-axis (for 2D problems).
Lt : float, default=1.0
Total simulation time.
Nt : int, default=100
Number of time steps.
initial_condition : callable
Function returning the initial state u(x, 0) or u(x, y, 0).
initial_velocity : callable, optional
Function returning the initial time derivative ∂ₜu(x, 0) or ∂ₜu(x, y, 0),
required for second-order equations.
n_frames : int, default=100
Number of time frames to store during simulation for visualization or output.
Raises
------
ValueError
If mandatory parameters are missing (e.g., Nx not given in 1D, Ly/Ny not given in 2D).
Notes
-----
- The spatial discretization assumes periodic boundary conditions by default.
- Fourier transforms are computed using real-to-complex FFTs (`scipy.fft.fft`, `fft2`).
- Frequency arrays (`KX`, `KY`) are defined following standard spectral conventions.
- Dealiasing is applied using a sharp cutoff filter at a fraction of the maximum frequency.
- For second-order equations, initial acceleration is derived from the governing operator.
- Symbolic analysis includes plotting of the symbol's real/imaginary/absolute values,
wavefront propagation, and dispersion relation.
See Also
--------
setup_1D : Sets up internal variables for one-dimensional problems.
setup_2D : Sets up internal variables for two-dimensional problems.
initialize_conditions : Applies initial data and enforces compatibility.
check_cfl_condition : Verifies time step against stability constraints.
plot_symbol : Visualizes the linear operator’s symbol in frequency space.
analyze_wave_propagation : Analyzes group velocity and wavefront dynamics.
interactive_symbol_analysis : Interactive tools for ψOp-based equations.
"""
# Temporal parameters
self.Lt, self.Nt = Lt, Nt
self.dt = Lt / Nt
self.n_frames = n_frames
self.frames = []
self.initial_condition = initial_condition
self.boundary_condition = boundary_condition
if self.boundary_condition == 'dirichlet' and not self.has_psi:
raise ValueError(
"Dirichlet boundary conditions require the equation to be defined via a pseudo-differential operator (psiOp). "
"Please provide an equation involving psiOp for non-periodic boundary treatment."
)
# Dimension checks
if self.dim == 1:
if Nx is None:
raise ValueError("Nx must be specified in 1D.")
self.setup_1D(Lx, Nx)
else:
if None in (Ly, Ny):
raise ValueError("In 2D, Ly and Ny must be provided.")
self.setup_2D(Lx, Ly, Nx, Ny)
# Initialization of solution and velocities
if not self.is_stationary:
self.initialize_conditions(initial_condition, initial_velocity)
# Symbol analysis if present
if self.has_psi:
print("⚠️ For psiOp, use interactive_symbol_analysis.")
else:
if self.L_symbolic == 0:
print("⚠️ Linear operator is null.")
else:
self.check_cfl_condition()
self.check_symbol_conditions()
self.plot_symbol()
if self.temporal_order == 2:
self.analyze_wave_propagation()
def setup_1D(self, Lx, Nx):
"""
Configure internal variables for one-dimensional (1D) problems.
This private method initializes spatial and frequency grids, applies dealiasing,
and prepares either pseudo-differential symbols or linear operators for use in time evolution.
It assumes periodic boundary conditions and uses real-to-complex FFT conventions.
The spatial domain is centered at zero: [-Lx/2, Lx/2].
Parameters
----------
Lx : float
Physical size of the spatial domain along the x-axis.
Nx : int
Number of grid points in the x-direction.
Attributes Set
--------------
- self.Lx : float
Size of the spatial domain.
- self.Nx : int
Number of spatial points.
- self.x_grid : np.ndarray
1D array of spatial coordinates.
- self.X : np.ndarray
Alias to `self.x_grid`, used in physical space computations.
- self.kx : np.ndarray
Array of wavenumbers corresponding to the Fourier transform.
- self.KX : np.ndarray
Alias to `self.kx`, used in frequency space computations.
- self.dealiasing_mask : np.ndarray
Boolean mask used to suppress aliased frequencies during nonlinear calculations.
- self.exp_L : np.ndarray
Exponential of the linear operator scaled by time step: exp(L(k) · dt).
- self.omega_val : np.ndarray
Frequency values ω(k) = Re[√(L(k))] used in second-order time stepping.
- self.cos_omega_dt, self.sin_omega_dt : np.ndarray
Cosine and sine of ω(k)·dt for dispersive propagation.
- self.inv_omega : np.ndarray
Inverse of ω(k), used to avoid division-by-zero in time stepping.
Notes
-----
- Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency.
- Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`.
- If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`.
- For second-order equations, the dispersion relation ω(k) is extracted from the linear operator L(k).
See Also
--------
setup_2D : Equivalent setup for two-dimensional problems.
prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation.
setup_omega_terms : Sets up terms involving ω(k) for second-order evolution.
"""
self.Lx, self.Nx = Lx, Nx
self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False)
self.X = self.x_grid
self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx)
self.KX = self.kx
# Dealiasing mask
k_max = self.dealiasing_ratio * np.max(np.abs(self.kx))
self.dealiasing_mask = (np.abs(self.KX) <= k_max)
# Preparation of symbol or linear operator
if self.has_psi:
self.prepare_symbol_tables()
else:
L_vals = np.array(self.L(self.KX), dtype=np.complex128)
self.exp_L = np.exp(L_vals * self.dt)
if self.temporal_order == 2:
omega_val = self.omega(self.KX)
self.setup_omega_terms(omega_val)
def setup_2D(self, Lx, Ly, Nx, Ny):
"""
Configure internal variables for two-dimensional (2D) problems.
This private method initializes spatial and frequency grids, applies dealiasing,
and prepares either pseudo-differential symbols or linear operators for use in time evolution.
It assumes periodic boundary conditions and uses real-to-complex FFT conventions.
The spatial domain is centered at zero: [-Lx/2, Lx/2] × [-Ly/2, Ly/2].
Parameters
----------
Lx : float
Physical size of the spatial domain along the x-axis.
Ly : float
Physical size of the spatial domain along the y-axis.
Nx : int
Number of grid points along the x-direction.
Ny : int
Number of grid points along the y-direction.
Attributes Set
--------------
- self.Lx, self.Ly : float
Size of the spatial domain in each direction.
- self.Nx, self.Ny : int
Number of spatial points in each direction.
- self.x_grid, self.y_grid : np.ndarray
1D arrays of spatial coordinates in x and y directions.
- self.X, self.Y : np.ndarray
2D meshgrids of spatial coordinates for physical space computations.
- self.kx, self.ky : np.ndarray
Arrays of wavenumbers corresponding to Fourier transforms in x and y directions.
- self.KX, self.KY : np.ndarray
Meshgrids of wavenumbers used in frequency space computations.
- self.dealiasing_mask : np.ndarray
Boolean mask used to suppress aliased frequencies during nonlinear calculations.
- self.exp_L : np.ndarray
Exponential of the linear operator scaled by time step: exp(L(kx, ky) · dt).
- self.omega_val : np.ndarray
Frequency values ω(kx, ky) = Re[√(L(kx, ky))] used in second-order time stepping.
- self.cos_omega_dt, self.sin_omega_dt : np.ndarray
Cosine and sine of ω(kx, ky)·dt for dispersive propagation.
- self.inv_omega : np.ndarray
Inverse of ω(kx, ky), used to avoid division-by-zero in time stepping.
Notes
-----
- Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency.
- Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`.
- If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`.
- For second-order equations, the dispersion relation ω(kx, ky) is extracted from the linear operator L(kx, ky).
See Also
--------
setup_1D : Equivalent setup for one-dimensional problems.
prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation.
setup_omega_terms : Sets up terms involving ω(kx, ky) for second-order evolution.
"""
self.Lx, self.Ly = Lx, Ly
self.Nx, self.Ny = Nx, Ny
self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False)
self.y_grid = np.linspace(-Ly/2, Ly/2, Ny, endpoint=False)
self.X, self.Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij')
self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx)
self.ky = 2 * np.pi * fftfreq(Ny, d=Ly / Ny)
self.KX, self.KY = np.meshgrid(self.kx, self.ky, indexing='ij')
# Dealiasing mask
kx_max = self.dealiasing_ratio * np.max(np.abs(self.kx))
ky_max = self.dealiasing_ratio * np.max(np.abs(self.ky))
self.dealiasing_mask = (np.abs(self.KX) <= kx_max) & (np.abs(self.KY) <= ky_max)
# Preparation of symbol or linear operator
if self.has_psi:
self.prepare_symbol_tables()
else:
L_vals = self.L(self.KX, self.KY)
self.exp_L = np.exp(L_vals * self.dt)
if self.temporal_order == 2:
omega_val = self.omega(self.KX, self.KY)
self.setup_omega_terms(omega_val)
def setup_omega_terms(self, omega_val):
"""
Initialize terms derived from the angular frequency ω for time evolution.
This private method precomputes and stores key trigonometric and inverse quantities
based on the dispersion relation ω(k), used in second-order time integration schemes.
These values are essential for solving wave-like equations with dispersive behavior:
cos(ω·dt), sin(ω·dt), 1/ω
The inverse frequency is computed safely to avoid division by zero.
Parameters
----------
omega_val : np.ndarray
Array of angular frequency values ω(k) evaluated at discrete wavenumbers.
Can be one-dimensional (1D) or two-dimensional (2D) depending on spatial dimension.
Attributes Set
--------------
- self.omega_val : np.ndarray
Copy of the input angular frequency array.
- self.cos_omega_dt : np.ndarray
Cosine of ω(k) multiplied by time step: cos(ω(k) · dt).
- self.sin_omega_dt : np.ndarray
Sine of ω(k) multiplied by time step: sin(ω(k) · dt).
- self.inv_omega : np.ndarray
Inverse of ω(k), with zeros where ω(k) == 0 to avoid division by zero.
Notes
-----
- This method is typically called during setup when solving second-order PDEs
involving dispersive waves (e.g., Klein-Gordon, Schrödinger, or water wave equations).
- The safe computation of 1/ω ensures numerical stability even when low frequencies are present.
- These precomputed arrays are used in spectral propagators for accurate time stepping.
See Also
--------
setup_1D : Sets up internal variables for one-dimensional problems.
setup_2D : Sets up internal variables for two-dimensional problems.
solve : Time integration using the computed frequency terms.
"""
self.omega_val = omega_val
self.cos_omega_dt = np.cos(omega_val * self.dt)
self.sin_omega_dt = np.sin(omega_val * self.dt)
self.inv_omega = np.zeros_like(omega_val)
nonzero = omega_val != 0
self.inv_omega[nonzero] = 1.0 / omega_val[nonzero]
def evaluate_source_at_t0(self):
"""
Evaluate source terms at initial time t = 0 over the spatial grid.
This private method computes the total contribution of all source terms at the initial time,
evaluated across the entire spatial domain. It supports both one-dimensional (1D) and
two-dimensional (2D) configurations.
Returns
-------
np.ndarray
A numpy array representing the evaluated source term at t=0:
- In 1D: Shape (Nx,), evaluated at each x in `self.x_grid`.
- In 2D: Shape (Nx, Ny), evaluated at each (x, y) pair in the grid.
Notes
-----
- The symbolic expressions in `self.source_terms` are substituted with numerical values at t=0.
- In 1D, each term is evaluated at (t=0, x=x_val).
- In 2D, each term is evaluated at (t=0, x=x_val, y=y_val).
- Evaluated using SymPy's `evalf()` to ensure numeric conversion.
- This method assumes that the source terms have already been lambdified or are compatible with symbolic substitution.
See Also
--------
setup : Initializes the spatial grid and source terms.
solve : Uses this evaluation during the first time step.
"""
if self.dim == 1:
# Evaluation on the 1D spatial grid
return np.array([
sum(term.subs(self.t, 0).subs(self.x, x_val).evalf()
for term in self.source_terms)
for x_val in self.x_grid
], dtype=np.float64)
else:
# Evaluation on the 2D spatial grid
return np.array([
[sum(term.subs({self.t: 0, self.x: x_val, self.y: y_val}).evalf()
for term in self.source_terms)
for y_val in self.y_grid]
for x_val in self.x_grid
], dtype=np.float64)
def initialize_conditions(self, initial_condition, initial_velocity):
"""
Initialize the solution and velocity fields at t = 0.
This private method sets up the initial state of the solution `u_prev` and, if applicable,
the time derivative (velocity) `v_prev` for second-order evolution equations.
For second-order equations, it also computes the backward-in-time value `u_prev2`
needed by the Leap-Frog method. The acceleration at t = 0 is computed from:
∂ₜ²u = L(u) + N(u) + f(x, t=0)
where L is the linear operator, N is the nonlinear term, and f is the source term.
Parameters
----------
initial_condition : callable
Function returning the initial condition u(x, 0) or u(x, y, 0).
initial_velocity : callable or None
Function returning the initial velocity ∂ₜu(x, 0) or ∂ₜu(x, y, 0). Required for
second-order equations; ignored otherwise.
Raises
------
ValueError
If `initial_velocity` is not provided for second-order equations.
Notes
-----
- Applies periodic boundary conditions after setting initial data.
- Stores a copy of the initial state in `self.frames` for visualization/output.
- In second-order systems, initializes `self.u_prev2` using a Taylor expansion:
u_prev2 = u_prev - dt * v_prev + 0.5 * dt² * (∂ₜ²u)
See Also
--------
apply_boundary : Enforces periodic boundary conditions on the solution field.
psiOp_apply : Computes pseudo-differential operator action for acceleration.
linear_rhs : Evaluates linear part of the equation in Fourier space.
apply_nonlinear : Handles nonlinear terms with spectral differentiation.
evaluate_source_at_t0 : Evaluates source terms at the initial time.
"""
# Initial condition
if self.dim == 1:
self.u_prev = initial_condition(self.X)
else:
self.u_prev = initial_condition(self.X, self.Y)
self.apply_boundary(self.u_prev)
# Initial velocity (second order)
if self.temporal_order == 2:
if initial_velocity is None:
raise ValueError("Initial velocity is required for second-order equations.")
if self.dim == 1:
self.v_prev = initial_velocity(self.X)
else:
self.v_prev = initial_velocity(self.X, self.Y)
self.u0 = np.copy(self.u_prev)
self.v0 = np.copy(self.v_prev)
# Calculation of u_prev2 (initial acceleration)
if not hasattr(self, 'u_prev2'):
if self.has_psi:
acc0 = self.apply_psiOp(self.u_prev)
else:
acc0 = self.linear_rhs(self.u_prev, is_v=False)
rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False)
acc0 += rhs_nl
if hasattr(self, 'source_terms') and self.source_terms:
acc0 += self.evaluate_source_at_t0()
self.u_prev2 = self.u_prev - self.dt * self.v_prev + 0.5 * self.dt**2 * acc0
self.frames = [self.u_prev.copy()]
def apply_boundary(self, u):
"""
Apply boundary conditions to the solution array based on the specified type.
This method supports two types of boundary conditions:
- 'periodic': Enforces periodicity by copying opposite boundary values.
- 'dirichlet': Sets all boundary values to zero (homogeneous Dirichlet condition).
Parameters
----------
u : np.ndarray
The solution array representing the field values on a spatial grid.
In 1D, shape must be (Nx,). In 2D, shape must be (Nx, Ny).
Raises
------
ValueError
If `self.boundary_condition` is not one of {'periodic', 'dirichlet'}.
Notes
-----
- For 'periodic':
* In 1D: u[0] = u[-2], u[-1] = u[1]
* In 2D: First and last rows/columns are set equal to their neighbors.
- For 'dirichlet':
* All boundary points are explicitly set to zero.
"""
if self.boundary_condition == 'periodic':
if self.dim == 1:
u[0] = u[-2]
u[-1] = u[1]
elif self.dim == 2:
u[0, :] = u[-2, :]
u[-1, :] = u[1, :]
u[:, 0] = u[:, -2]
u[:, -1] = u[:, 1]
elif self.boundary_condition == 'dirichlet':
if self.dim == 1:
u[0] = 0
u[-1] = 0
elif self.dim == 2:
u[0, :] = 0
u[-1, :] = 0
u[:, 0] = 0
u[:, -1] = 0
else:
raise ValueError(
f"Invalid boundary condition '{self.boundary_condition}'. "
"Supported types are 'periodic' and 'dirichlet'."
)
def apply_nonlinear(self, u, is_v=False):
"""
Apply nonlinear terms to the solution using spectral differentiation with dealiasing.
This method evaluates all nonlinear terms present in the PDE by substituting spatial
derivatives with their spectral approximations computed via FFT. The dealiasing mask
ensures numerical stability by removing high-frequency components that could lead
to aliasing errors.
Parameters
----------
u : numpy.ndarray
Current solution array on the spatial grid.
is_v : bool
If True, evaluates nonlinear terms for the velocity field v instead of u.
Returns:
numpy.ndarray: Array representing the contribution of nonlinear terms multiplied by dt.
Notes:
- In 1D, computes ∂ₓu via FFT and substitutes any derivative term in the nonlinear expressions.
- In 2D, computes ∂ₓu and ∂ᵧu via FFT and performs similar substitutions.
- Uses lambdify to evaluate symbolic nonlinear expressions numerically.
- Derivatives are replaced symbolically with 'u_x' and 'u_y' before evaluation.
"""
if not self.nonlinear_terms:
return np.zeros_like(u, dtype=np.complex128)
nonlinear_term = np.zeros_like(u, dtype=np.complex128)
if self.dim == 1:
u_hat = self.fft(u)
u_hat *= self.dealiasing_mask
u = self.ifft(u_hat)
u_x_hat = (1j * self.KX) * u_hat
u_x = self.ifft(u_x_hat)
for term in self.nonlinear_terms:
term_replaced = term
if term.has(Derivative):
for deriv in term.atoms(Derivative):
if deriv.args[1][0] == self.x:
term_replaced = term_replaced.subs(deriv, symbols('u_x'))
term_func = lambdify((self.t, self.x, self.u_eq, 'u_x'), term_replaced, 'numpy')
if is_v:
nonlinear_term += term_func(0, self.X, self.v_prev, u_x)
else:
nonlinear_term += term_func(0, self.X, u, u_x)
elif self.dim == 2:
u_hat = self.fft(u)
u_hat *= self.dealiasing_mask
u = self.ifft(u_hat)
u_x_hat = (1j * self.KX) * u_hat
u_y_hat = (1j * self.KY) * u_hat
u_x = self.ifft(u_x_hat)
u_y = self.ifft(u_y_hat)
for term in self.nonlinear_terms:
term_replaced = term
if term.has(Derivative):
for deriv in term.atoms(Derivative):
if deriv.args[1][0] == self.x:
term_replaced = term_replaced.subs(deriv, symbols('u_x'))
elif deriv.args[1][0] == self.y:
term_replaced = term_replaced.subs(deriv, symbols('u_y'))
term_func = lambdify((self.t, self.x, self.y, self.u_eq, 'u_x', 'u_y'), term_replaced, 'numpy')
if is_v:
nonlinear_term += term_func(0, self.X, self.Y, self.v_prev, u_x, u_y)
else:
nonlinear_term += term_func(0, self.X, self.Y, u, u_x, u_y)
else:
raise ValueError("Unsupported spatial dimension.")
return nonlinear_term * self.dt
def prepare_symbol_tables(self):
"""
Precompute and store evaluated pseudo-differential operator symbols for spectral methods.
This method evaluates all pseudo-differential operators (ψOp) present in the PDE
over the spatial and frequency grids, scales them by their respective coefficients,
and combines them into a single composite symbol used in time-stepping and inversion.
The evaluation is performed via the `evaluate` method of each PseudoDifferentialOperator,
which computes p(x, ξ) or p(x, y, ξ, η) numerically over the current grid configuration.
Side Effects:
self.precomputed_symbols : list of (coeff, symbol_array)
Each tuple contains a coefficient and its evaluated symbol on the grid.
self.combined_symbol : np.ndarray
Sum of all scaled symbol arrays: ∑(coeffₖ * ψₖ(x, ξ))
Raises:
ValueError: If the spatial dimension is not 1D or 2D.
"""
self.precomputed_symbols = []
self.combined_symbol = 0
for coeff, psi in self.psi_ops:
if self.dim == 1:
raw = psi.evaluate(self.X, None, self.KX, None)
elif self.dim == 2:
raw = psi.evaluate(self.X, self.Y, self.KX, self.KY)
else:
raise ValueError('Unsupported spatial dimension.')
raw_flat = raw.flatten()
converted = np.array([complex(N(val)) for val in raw_flat], dtype=np.complex128)
raw_eval = converted.reshape(raw.shape)
self.precomputed_symbols.append((coeff, raw_eval))
self.combined_symbol = sum((coeff * sym for coeff, sym in self.precomputed_symbols))
self.combined_symbol = np.array(self.combined_symbol, dtype=np.complex128)
def total_symbol_expr(self):
"""
Compute the total pseudo-differential symbol expression from all pseudo_terms.
This method constructs the full symbol of the pseudo-differential operator
by summing up all coefficient-weighted symbolic expressions.
The result is cached in self.symbol_expr to avoid recomputation.
Returns:
sympy.Expr: The combined symbol expression, representing the full
pseudo-differential operator in symbolic form.
Example:
Given pseudo_terms = [(2, ξ²), (1, x·ξ)], this returns 2·ξ² + x·ξ.
"""
if not hasattr(self, '_symbol_expr'):
self.symbol_expr = sum(coeff * expr for coeff, expr in self.pseudo_terms)
return self.symbol_expr
def build_symbol_func(self, expr):
"""
Build a numerical evaluation function from a symbolic pseudo-differential operator expression.
This method converts a symbolic expression representing a pseudo-differential operator into
a callable NumPy-compatible function. The function accepts spatial and frequency variables
depending on the dimensionality of the problem.
Parameters
----------
expr : sympy expression
A SymPy expression representing the symbol of the pseudo-differential operator. It may depend on spatial variables (x, y) and frequency variables (xi, eta).
Returns:
function : A lambdified function that takes:
- In 1D: `(x, xi)` — spatial coordinate and frequency.
- In 2D: `(x, y, xi, eta)` — spatial coordinates and frequencies.
Returns a NumPy array of evaluated symbol values over input grids.
Notes:
- Uses `lambdify` from SymPy with the `'numpy'` backend for efficient vectorized evaluation.
- Real variable assumptions are enforced to ensure proper behavior in numerical contexts.
- Used internally by methods like `apply_psiOp`, `evaluate`, and visualization tools.
"""
if self.dim == 1:
x, xi = symbols('x xi', real=True)
return lambdify((x, xi), expr, 'numpy')
else:
x, y, xi, eta = symbols('x y xi eta', real=True)
return lambdify((x, y, xi, eta), expr, 'numpy')
def apply_psiOp(self, u):
"""
Apply the pseudo-differential operator to the input field u.
This method dispatches the application of the pseudo-differential operator based on:
- Whether the symbol is spatially dependent (x/y)
- The boundary condition in use (periodic or dirichlet)
Supported operations:
- Constant-coefficient symbols: applied via Fourier multiplication.
- Spatially varying symbols: applied via Kohn–Nirenberg quantization.
- Dirichlet boundary conditions: handled with non-periodic convolution-like quantization.
Dispatch Logic:\n
if not self.is_spatial: u ↦ Op(p)(D) ⋅ u = 𝓕⁻¹[ p(ξ) ⋅ 𝓕(u) ]\n
elif periodic: u ↦ Op(p)(x,D) ⋅ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ based of FFT (quicker)\n
elif dirichlet: u ↦ Op(p)(x,D) ⋅ u ≈ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ (slower)\n
Parameters
----------
u : np.ndarray
Input field to which the operator is applied.
Should be 1D or 2D depending on the problem dimension.
Returns:
np.ndarray: Result of applying the pseudo-differential operator to u.
Raises:
ValueError: If an unsupported boundary condition is specified.
"""
if not self.is_spatial:
return self.apply_psiOp_constant(u)
elif self.boundary_condition == 'periodic':
return self.apply_psiOp_kohn_nirenberg_fft(u)
elif self.boundary_condition == 'dirichlet':
return self.apply_psiOp_kohn_nirenberg_nonperiodic(u)
else:
raise ValueError(f"Invalid boundary condition '{self.boundary_condition}'")
def apply_psiOp_constant(self, u):
"""
Apply a constant-coefficient pseudo-differential operator in Fourier space.
This method assumes the symbol is diagonal in the Fourier basis and acts as a
multiplication operator. It performs the operation:
(ψu)(x) = 𝓕⁻¹[ -σ(k) · 𝓕[u](k) ]
where:
- σ(k) is the combined pseudo-differential operator symbol
- 𝓕 denotes the forward Fourier transform
- 𝓕⁻¹ denotes the inverse Fourier transform
The dealiasing mask is applied before returning to physical space.
Parameters
----------
u : np.ndarray
Input function in physical space (real-valued or complex-valued)
Returns:
np.ndarray : Result of applying the pseudo-differential operator to u, same shape as input
"""
u_hat = self.fft(u)
u_hat *= -self.combined_symbol
u_hat *= self.dealiasing_mask
return self.ifft(u_hat)
def apply_psiOp_kohn_nirenberg_fft(self, u):
"""
Apply a pseudo-differential operator using the Kohn–Nirenberg quantization in Fourier space.
This method evaluates the action of a pseudo-differential operator defined by the total symbol,
computed from all psiOp terms in the equation. It uses the fast Fourier transform (FFT) for
efficiency in periodic domains.
Parameters
----------
u : np.ndarray
Input function in real space to which the operator is applied.
Returns:
np.ndarray: Resulting function after applying the pseudo-differential operator.
Process:
1. Compute the total symbolic expression of the pseudo-differential operator.
2. Build a callable numerical function from the symbol.
3. Evaluate Op(p)(u) via the Kohn–Nirenberg quantization using FFT.
Note:
- Assumes periodic boundary conditions.
- The returned result is the negative of the standard definition due to PDE sign conventions.
"""
total_symbol = self.total_symbol_expr()
symbol_func = self.build_symbol_func(total_symbol)
return -self.kohn_nirenberg_fft(u_vals=u, symbol_func=symbol_func)
def apply_psiOp_kohn_nirenberg_nonperiodic(self, u):
"""
Apply a pseudo-differential operator using the Kohn–Nirenberg quantization on non-periodic domains.
This method evaluates the action of a pseudo-differential operator Op(p) on a function u
via the Kohn–Nirenberg representation. It supports both 1D and 2D cases and uses spatial
and frequency grids to evaluate the operator symbol p(x, ξ).
The operator symbol p(x, ξ) is extracted from the PDE and evaluated numerically using
`_total_symbol_expr` and `_build_symbol_func`.
Parameters
----------
u : np.ndarray
Input function (real space) to which the operator is applied.
Returns:
np.ndarray: Result of applying Op(p) to u in real space.
Notes:
- For 1D: p(x, ξ) is evaluated over x_grid and xi_grid.
- For 2D: p(x, y, ξ, η) is evaluated over (x_grid, y_grid) and (xi_grid, eta_grid).
- The result is computed using `kohn_nirenberg_nonperiodic`, which handles non-periodic boundary conditions.
"""
total_symbol = self.total_symbol_expr()
symbol_func = self.build_symbol_func(total_symbol)
if self.dim == 1:
return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=self.x_grid, xi_grid=self.kx, symbol_func=symbol_func)
else:
return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=symbol_func)
def step_order1_with_psi(self, source_contribution):
"""
Perform one time step of a first-order evolution using a pseudo-differential operator.
This method updates the solution field using an exponential integrator or explicit Euler scheme,
depending on boundary conditions and the structure of the pseudo-differential symbol.
It supports:
- Linear dynamics via pseudo-differential operator L (possibly nonlocal)
- Nonlinear terms computed via spectral differentiation
- External source contributions
The update follows **three distinct computational paths**:
1. **Periodic boundaries + diagonalizable symbol**
Symbol is constant in space → use direct Fourier-based exponential integrator:
uₙ₊₁ = e⁻ᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(−LΔt) ⋅ (N(uₙ) + F)
2. **Non-diagonalizable but spatially uniform symbol**
General exponential time differencing of order 1:
uₙ₊₁ = eᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(LΔt) ⋅ (N(uₙ) + F)
3. **Spatially varying symbol**
No frequency diagonalization available → use explicit Euler:
uₙ₊₁ = uₙ + Δt ⋅ (L(uₙ) + N(uₙ) + F)
where:
L(uₙ) = linear part via pseudo-differential operator
N(uₙ) = nonlinear contribution at current time step
F = external source term
Δt = time step size
φ₁(z) = (eᶻ − 1)/z (with safe handling near z=0)
Boundary conditions are applied after each update to ensure consistency.
Parameters
source_contribution (np.ndarray): Array representing the external source term at current time step.
Must match the spatial dimensions of self.u_prev.
Returns:
np.ndarray: Updated solution array after one time step.
"""
# Handling null source
if np.isscalar(source_contribution):
source = np.zeros_like(self.u_prev)
else:
source = source_contribution
def spectral_filter(u, cutoff=0.8):
if u.ndim == 1:
u_hat = self.fft(u)
N = len(u)
k = fftfreq(N)
mask = np.exp(-(k / cutoff)**8)
return self.ifft(u_hat * mask).real
elif u.ndim == 2:
u_hat = self.fft(u)
Ny, Nx = u.shape
ky = fftfreq(Ny)[:, None]
kx = fftfreq(Nx)[None, :]
k_squared = kx**2 + ky**2
mask = np.exp(-(np.sqrt(k_squared) / cutoff)**8)
return self.ifft(u_hat * mask).real
else:
raise ValueError("Only 1D and 2D arrays are supported.")
# Recalculate symbol if necessary
if self.is_spatial:
self.prepare_symbol_tables() # Recalculates self.combined_symbol
# Case with FFT (symbol diagonalizable in Fourier space)
if self.boundary_condition == 'periodic' and not self.is_spatial:
u_hat = self.fft(self.u_prev)
u_hat *= np.exp(-self.dt * self.combined_symbol)
u_hat *= self.dealiasing_mask
u_symb = self.ifft(u_hat)
u_nl = self.apply_nonlinear(self.u_prev)
u_new = u_symb + u_nl + source
else:
if not self.is_spatial:
# General case with ETD1
u_nl = self.apply_nonlinear(self.u_prev)
# Calculation of exp(dt * L) and phi1(dt * L)
L_vals = self.combined_symbol # Uses the updated symbol
exp_L = np.exp(-self.dt * L_vals)
phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
phi1_L[np.isnan(phi1_L)] = 1.0 # Handling division by zero
# Fourier transform
u_hat = self.fft(self.u_prev)
u_nl_hat = self.fft(u_nl)
source_hat = self.fft(source)
# Assembling the solution in Fourier space
u_hat_new = exp_L * u_hat + self.dt * phi1_L * (u_nl_hat + source_hat)
u_new = self.ifft(u_hat_new)
else:
# if the symbol depends on spatial variables : Euler method
Lu_prev = self.apply_psiOp(self.u_prev)
u_nl = self.apply_nonlinear(self.u_prev)
u_new = self.u_prev + self.dt * (Lu_prev + u_nl + source)
u_new = spectral_filter(u_new, cutoff=self.dealiasing_ratio)
# Applying boundary conditions
self.apply_boundary(u_new)
return u_new
def step_order2_with_psi(self, source_contribution):
"""
Perform one time step of a second-order time evolution using a pseudo-differential operator.
This method updates the solution field using a second-order accurate scheme suitable for wave-like equations.
The update includes contributions from:
- Linear dynamics via a pseudo-differential operator (e.g., dispersion or stiffness)
- Nonlinear terms computed via spectral differentiation
- External source contributions
Discretization follows a leapfrog-style finite difference in time:
uₙ₊₁ = 2uₙ − uₙ₋₁ + Δt² ⋅ (L(uₙ) + N(uₙ) + F)
where:
L(uₙ) = linear part evaluated via pseudo-differential operator
N(uₙ) = nonlinear contribution at current time step
F = external source term at current time step
Δt = time step size
Boundary conditions are applied after each update to ensure consistency.
Parameters
source_contribution (np.ndarray): Array representing the external source term at current time step.
Must match the spatial dimensions of self.u_prev.
Returns:
np.ndarray: Updated solution array after one time step.
"""
Lu_prev = self.apply_psiOp(self.u_prev)
rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False)
u_new = 2 * self.u_prev - self.u_prev2 + self.dt ** 2 * (Lu_prev + rhs_nl + source_contribution)
self.apply_boundary(u_new)
self.u_prev2 = self.u_prev
self.u_prev = u_new
self.u = u_new
return u_new
def solve(self):
"""
Solve the partial differential equation numerically using spectral methods.
This method evolves the solution in time using a combination of:
- Fourier-based linear evolution (with dealiasing)
- Nonlinear term handling via pseudo-spectral evaluation
- Support for pseudo-differential operators (psiOp)
- Source terms and boundary conditions
The solver supports:
- 1D and 2D spatial domains
- First and second-order time evolution
- Periodic and Dirichlet boundary conditions
- Time-stepping schemes: default, ETD-RK4
Returns:
list[np.ndarray]: A list of solution arrays at each saved time frame.
Side Effects:
- Updates self.frames: stores solution snapshots
- Updates self.energy_history: records total energy if enabled
Algorithm Overview:
For each time step:
1. Evaluate source contributions (if any)
2. Apply time evolution:
- Order 1:
- With psiOp: uses step_order1_with_psi
- With ETD-RK4: exponential time differencing
- Default: linear + nonlinear update
- Order 2:
- With psiOp: uses step_order2_with_psi
- With ETD-RK4: second-order exponential scheme
- Default: second-order leapfrog-style update
3. Enforce boundary conditions
4. Save solution snapshot periodically
5. Record energy (for second-order systems without psiOp)
"""
print('\n*******************')
print('* Solving the PDE *')
print('*******************\n')
save_interval = max(1, self.Nt // self.n_frames)
self.energy_history = []
for step in range(self.Nt):
if hasattr(self, 'source_terms') and self.source_terms:
source_contribution = np.zeros_like(self.X, dtype=np.float64)
for term in self.source_terms:
try:
if self.dim == 1:
source_func = lambdify((self.t, self.x), term, 'numpy')
source_contribution += source_func(step * self.dt, self.X)
elif self.dim == 2:
source_func = lambdify((self.t, self.x, self.y), term, 'numpy')
source_contribution += source_func(step * self.dt, self.X, self.Y)
except Exception as e:
print(f'Error evaluating source term {term}: {e}')
else:
source_contribution = 0
if self.temporal_order == 1:
if self.has_psi:
u_new = self.step_order1_with_psi(source_contribution)
elif hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4':
u_new = self.step_ETD_RK4(self.u_prev)
else:
u_hat = self.fft(self.u_prev)
u_hat *= self.exp_L
u_hat *= self.dealiasing_mask
u_lin = self.ifft(u_hat)
u_nl = self.apply_nonlinear(u_lin)
u_new = u_lin + u_nl + source_contribution
self.apply_boundary(u_new)
self.u_prev = u_new
elif self.temporal_order == 2:
if self.has_psi:
u_new = self.step_order2_with_psi(source_contribution)
else:
if hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4':
u_new, v_new = self.step_ETD_RK4_order2(self.u_prev, self.v_prev)
else:
u_hat = self.fft(self.u_prev)
v_hat = self.fft(self.v_prev)
u_new_hat = self.cos_omega_dt * u_hat + self.sin_omega_dt * self.inv_omega * v_hat
v_new_hat = -self.omega_val * self.sin_omega_dt * u_hat + self.cos_omega_dt * v_hat
u_new = self.ifft(u_new_hat)
v_new = self.ifft(v_new_hat)
u_nl = self.apply_nonlinear(self.u_prev, is_v=False)
v_nl = self.apply_nonlinear(self.v_prev, is_v=True)
u_new += (u_nl + source_contribution) * self.dt ** 2 / 2
v_new += (u_nl + source_contribution) * self.dt
self.apply_boundary(u_new)
self.apply_boundary(v_new)
self.u_prev = u_new
self.v_prev = v_new
if step % save_interval == 0:
self.frames.append(self.u_prev.copy())
if self.temporal_order == 2 and (not self.has_psi):
E = self.compute_energy()
self.energy_history.append(E)
return self.frames
def solve_stationary_psiOp(self, order=3):
"""
Solve stationary pseudo-differential equations of the form P[u] = f(x) or P[u] = f(x,y) using asymptotic inversion.
This method computes the solution to a stationary (time-independent) pseudo-differential equation
where the operator P is defined via symbolic expressions (psiOp). It constructs an asymptotic right inverse R
such that P∘R ≈ Id, then applies it to the source term f using either direct Fourier multiplication
(when the symbol is spatially independent) or Kohn–Nirenberg quantization (when spatial dependence is present).
The inversion is based on the principal symbol of the operator and its asymptotic expansion up to the given order.
Ellipticity of the symbol is checked numerically before inversion to ensure well-posedness.
Parameters
----------
order : int, default=3
Order of the asymptotic expansion used to construct the right inverse of the pseudo-differential operator.
method : str, optional
Inversion strategy:
- 'diagonal' (default): Fast approximate inversion using diagonal operators in frequency space.
- 'full' : Pointwise exact inversion (slower but more accurate).
Returns
-------
ndarray
The computed solution u(x) in 1D or u(x, y) in 2D as a NumPy array over the spatial grid.
Raises
------
ValueError
If no pseudo-differential operator (psiOp) is defined.
If linear or nonlinear terms other than psiOp are present.
If the symbol is not elliptic on the grid.
If no source term is provided for the right-hand side.
Notes
-----
- The method assumes the problem is fully stationary: time derivatives must be absent.
- Requires the equation to be purely pseudo-differential (no Op, Derivative, or nonlinear terms).
- Symbol evaluation and inversion are dimension-aware (supports both 1D and 2D problems).
- Supports optimization paths when the symbol does not depend on spatial variables.
See Also
--------
right_inverse_asymptotic : Constructs the asymptotic inverse of the pseudo-differential operator.
kohn_nirenberg : Numerical implementation of general pseudo-differential operators.
is_elliptic_numerically : Verifies numerical ellipticity of the symbol.
"""
print("\n*******************************")
print("* Solving the stationnary PDE *")
print("*******************************\n")
print("boundary condition: ",self.boundary_condition)
if not self.has_psi:
raise ValueError("Only supports problems with psiOp.")
if self.linear_terms or self.nonlinear_terms:
raise ValueError("Stationary psiOp problems must be linear and purely pseudo-differential.")
if self.boundary_condition not in ('periodic', 'dirichlet'):
raise ValueError(
"For stationary PDEs, boundary conditions must be explicitly defined. "
"Supported types are 'periodic' and 'dirichlet'."
)
if self.dim == 1:
x = self.x
xi = symbols('xi', real=True)
spatial_vars = (x,)
freq_vars = (xi,)
X, KX = self.X, self.KX
elif self.dim == 2:
x, y = self.x, self.y
xi, eta = symbols('xi eta', real=True)
spatial_vars = (x, y)
freq_vars = (xi, eta)
X, Y, KX, KY = self.X, self.Y, self.KX, self.KY
else:
raise ValueError("Unsupported spatial dimension.")
total_symbol = sum(coeff * psi.expr for coeff, psi in self.psi_ops)
psi_total = PseudoDifferentialOperator(total_symbol, spatial_vars, mode='symbol')
# Check ellipticity
if self.dim == 1:
is_elliptic = psi_total.is_elliptic_numerically(X, KX)
else:
is_elliptic = psi_total.is_elliptic_numerically((X[:, 0], Y[0, :]), (KX[:, 0], KY[0, :]))
if not is_elliptic:
raise ValueError("❌ The pseudo-differential symbol is not numerically elliptic on the grid.")
print("✅ Elliptic pseudo-differential symbol: inversion allowed.")
R_symbol = psi_total.right_inverse_asymptotic(order=order)
print("Right inverse asymptotic symbol:")
pprint(R_symbol, num_columns=150)
if self.dim == 1:
if R_symbol.has(x):
R_func = lambdify((x, xi), R_symbol, modules='numpy')
else:
R_func = lambdify((xi,), R_symbol, modules='numpy')
else:
if R_symbol.has(x) or R_symbol.has(y):
R_func = lambdify((x, y, xi, eta), R_symbol, modules='numpy')
else:
R_func = lambdify((xi, eta), R_symbol, modules='numpy')
# Build rhs
if self.source_terms:
f_expr = sum(self.source_terms)
used_vars = [v for v in spatial_vars if f_expr.has(v)]
f_func = lambdify(used_vars, -f_expr, modules='numpy')
if self.dim == 1:
rhs = f_func(self.x_grid) if used_vars else np.zeros_like(self.x_grid)
else:
rhs = f_func(self.X, self.Y) if used_vars else np.zeros_like(self.X)
elif self.initial_condition:
raise ValueError("Initial condition should be None for stationnary equation.")
else:
raise ValueError("No source term provided to construct the right-hand side.")
f_hat = self.fft(rhs)
if self.boundary_condition == 'periodic':
if self.dim == 1:
if not R_symbol.has(x):
print("⚡ Optimization: symbol independent of x — direct product in Fourier.")
R_vals = R_func(self.KX)
u_hat = R_vals * f_hat
u = self.ifft(u_hat)
else:
print("⚙️ 1D Kohn-Nirenberg Quantification")
x, xi = symbols('x xi', real=True)
R_func = lambdify((x, xi), R_symbol, 'numpy') # Still 2 args for uniformity
u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func)
elif self.dim == 2:
if not R_symbol.has(x) and not R_symbol.has(y):
print("⚡ Optimization: Symbol independent of x and y — direct product in 2D Fourier.")
R_vals = np.vectorize(R_func)(self.KX, self.KY)
u_hat = R_vals * f_hat
u = self.ifft(u_hat)
else:
print("⚙️ 2D Kohn-Nirenberg Quantification")
x, xi, y, eta = symbols('x xi y eta', real=True)
R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy') # Still 2 args for uniformity
u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func)
self.u = u
return u
elif self.boundary_condition == 'dirichlet':
if self.dim == 1:
x, xi = symbols('x xi', real=True)
R_func = lambdify((x, xi), R_symbol, 'numpy')
u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=X, xi_grid=KX, symbol_func=R_func)
elif self.dim == 2:
x, xi, y, eta = symbols('x xi y eta', real=True)
R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy')
u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=R_func)
self.u = u
return u
else:
raise ValueError(
f"Invalid boundary condition '{self.boundary_condition}'. "
"Supported types are 'periodic' and 'dirichlet'."
)
def kohn_nirenberg_fft(self, u_vals, symbol_func,
freq_window='gaussian', clamp=1e6,
space_window=False):
"""
Numerically stable Kohn–Nirenberg quantization of a pseudo-differential operator.
Applies the pseudo-differential operator Op(p) to the function f via the Kohn–Nirenberg quantization:
[Op(p)f](x) = (1/(2π)^d) ∫ p(x, ξ) e^{ix·ξ} ℱ[f](ξ) dξ
where p(x, ξ) is a symbol that may depend on both spatial variables x and frequency variables ξ.
This method supports both 1D and 2D cases and includes optional smoothing techniques to improve numerical stability.
Parameters
----------
u_vals : np.ndarray
Spatial samples of the input function f(x) or f(x, y), defined on a uniform grid.
symbol_func : callable
A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D.
Must accept NumPy-compatible array inputs and return a complex-valued array.
freq_window : {'gaussian', 'hann', None}, optional
Type of frequency-domain window to apply:
- 'gaussian': smooth decay near high frequencies
- 'hann': cosine-based tapering with hard cutoff
- None: no frequency window applied
clamp : float, optional
Upper bound on the absolute value of the symbol. Prevents numerical blow-up from large values.
space_window : bool, optional
Whether to apply a spatial Gaussian window to suppress edge effects in physical space.
Returns
-------
np.ndarray
The result of applying the pseudo-differential operator to f, returned as a real or complex array
of the same shape as u_vals.
Notes
-----
- The implementation uses FFT-based quadrature of the inverse Fourier transform.
- Symbol evaluation is vectorized over spatial and frequency grids.
- Frequency and spatial windows help mitigate oscillatory behavior and aliasing.
- In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η).
"""
# === Common setup ===
xg = self.x_grid
dx = xg[1] - xg[0]
if self.dim == 1:
# === 1D case ===
# Frequency grid (shifted to center zero)
Nx = self.Nx
k = 2 * np.pi * fftshift(fftfreq(Nx, d=dx))
dk = k[1] - k[0]
# Centered FFT of input
f_shift = fftshift(u_vals)
f_hat = self.fft(f_shift) * dx
f_hat = fftshift(f_hat)
# Build meshgrid for (x, ξ)
X, K = np.meshgrid(xg, k, indexing='ij')
# Evaluate the symbol p(x, ξ)
P = symbol_func(X, K)
# Optional: clamp extreme values
P = np.clip(P, -clamp, clamp)
# === Frequency-domain window ===
if freq_window == 'gaussian':
sigma = 0.8 * np.max(np.abs(k))
W = np.exp(-(K / sigma) ** 4)
P *= W
elif freq_window == 'hann':
W = 0.5 * (1 + np.cos(np.pi * K / np.max(np.abs(K))))
P *= W * (np.abs(K) < np.max(np.abs(K)))
# === Optional spatial window ===
if space_window:
x0 = (xg[0] + xg[-1]) / 2
L = (xg[-1] - xg[0]) / 2
S = np.exp(-((X - x0) / L) ** 2)
P *= S
# === Oscillatory kernel and integration ===
kernel = np.exp(1j * X * K)
integrand = P * f_hat[None, :] * kernel
# Approximate inverse Fourier integral
u = np.sum(integrand, axis=1) * dk / (2 * np.pi)
return u
else:
# === 2D case ===
yg = self.y_grid
dy = yg[1] - yg[0]
Nx, Ny = self.Nx, self.Ny
# Frequency grids
kx = 2 * np.pi * fftshift(fftfreq(Nx, d=dx))
ky = 2 * np.pi * fftshift(fftfreq(Ny, d=dy))
dkx = kx[1] - kx[0]
dky = ky[1] - ky[0]
# 2D FFT of f(x, y)
f_shift = fftshift(u_vals)
f_hat = self.fft(f_shift) * dx * dy
f_hat = fftshift(f_hat)
# Create 4D grids for broadcasting
X, Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij')
KX, KY = np.meshgrid(kx, ky, indexing='ij')
Xb = X[:, :, None, None]
Yb = Y[:, :, None, None]
KXb = KX[None, None, :, :]
KYb = KY[None, None, :, :]
# Evaluate p(x, y, ξ, η)
P_vals = symbol_func(Xb, Yb, KXb, KYb)
P_vals = np.clip(P_vals, -clamp, clamp)
# === Frequency windowing ===
if freq_window == 'gaussian':
sigma_kx = 0.8 * np.max(np.abs(kx))
sigma_ky = 0.8 * np.max(np.abs(ky))
W_kx = np.exp(-(KXb / sigma_kx) ** 4)
W_ky = np.exp(-(KYb / sigma_ky) ** 4)
P_vals *= W_kx * W_ky
elif freq_window == 'hann':
Wx = 0.5 * (1 + np.cos(np.pi * KXb / np.max(np.abs(kx))))
Wy = 0.5 * (1 + np.cos(np.pi * KYb / np.max(np.abs(ky))))
mask_x = np.abs(KXb) < np.max(np.abs(kx))
mask_y = np.abs(KYb) < np.max(np.abs(ky))
P_vals *= Wx * Wy * mask_x * mask_y
# === Optional spatial tapering ===
if space_window:
x0 = (self.x_grid[0] + self.x_grid[-1]) / 2
y0 = (self.y_grid[0] + self.y_grid[-1]) / 2
Lx = (self.x_grid[-1] - self.x_grid[0]) / 2
Ly = (self.y_grid[-1] - self.y_grid[0]) / 2
S = np.exp(-((Xb - x0) / Lx) ** 2 - ((Yb - y0) / Ly) ** 2)
P_vals *= S
# === Oscillatory kernel and integration ===
phase = np.exp(1j * (Xb * KXb + Yb * KYb))
integrand = P_vals * phase * f_hat[None, None, :, :]
# 2D Fourier inversion (numerical integration)
u = np.sum(integrand, axis=(2, 3)) * dkx * dky / (2 * np.pi) ** 2
return u
def kohn_nirenberg_nonperiodic(self, u_vals, x_grid, xi_grid, symbol_func,
freq_window='gaussian', clamp=1e6, space_window=False):
"""
Numerically applies the Kohn–Nirenberg quantization of a pseudo-differential operator
in a non-periodic setting.
This method computes:
[Op(p)u](x) = (1/(2π)^d) ∫ p(x, ξ) e^{i x·ξ} ℱ[u](ξ) dξ
where p(x, ξ) is a general symbol that may depend on both spatial and frequency variables.
It supports both 1D and 2D inputs and includes optional numerical smoothing techniques
to enhance stability for non-smooth or oscillatory symbols.
Parameters
----------
u_vals : np.ndarray
Input function values defined on a uniform spatial grid. Can be 1D (Nx,) or 2D (Nx, Ny).
x_grid : np.ndarray
Spatial grid points along each axis. In 1D: shape (Nx,). In 2D: tuple of two arrays (X, Y)
or list of coordinate arrays.
xi_grid : np.ndarray
Frequency grid points. In 1D: shape (Nxi,). In 2D: tuple of two arrays (Xi, Eta)
or list of frequency arrays.
symbol_func : callable
A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D.
Must accept NumPy-compatible array inputs and return a complex-valued array.
freq_window : {'gaussian', 'hann', None}, optional
Type of frequency-domain window to apply for regularization:
- 'gaussian': Smooth exponential decay near high frequencies.
- 'hann': Cosine-based tapering with hard cutoff.
- None: No frequency window applied.
clamp : float, optional
Maximum absolute value allowed for the symbol to prevent numerical overflow.
Default is 1e6.
space_window : bool, optional
If True, applies a smooth spatial Gaussian window centered in the domain to reduce
boundary artifacts. Default is False.
Returns
-------
np.ndarray
The result of applying the pseudo-differential operator Op(p) to u. Shape matches u_vals.
Notes
-----
- This version does not assume periodicity and is suitable for Dirichlet or Neumann boundary conditions.
- In 1D, the integral is evaluated as a sum over (x, ξ), using matrix exponentials.
- In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η), which can be computationally intensive.
- Symbol evaluation should be vectorized for performance.
- For large grids, consider reducing resolution via resampling before calling this function.
See Also
--------
kohn_nirenberg_fft : Faster implementation for periodic domains using FFT.
PseudoDifferentialOperator : Class for symbolic manipulation of pseudo-differential operators.
"""
if u_vals.ndim == 1:
# === 1D case ===
x = x_grid
xi = xi_grid
dx = x[1] - x[0]
dxi = xi[1] - xi[0]
phase_ft = np.exp(-1j * np.outer(xi, x)) # (Nxi, Nx)
u_hat = dx * np.dot(phase_ft, u_vals) # (Nxi,)
X, XI = np.meshgrid(x, xi, indexing='ij') # (Nx, Nxi)
sigma_vals = symbol_func(X, XI)
# Clamp values
sigma_vals = np.clip(sigma_vals, -clamp, clamp)
# Frequency window
if freq_window == 'gaussian':
sigma = 0.8 * np.max(np.abs(XI))
window = np.exp(-(XI / sigma)**4)
sigma_vals *= window
elif freq_window == 'hann':
window = 0.5 * (1 + np.cos(np.pi * XI / np.max(np.abs(XI))))
sigma_vals *= window * (np.abs(XI) < np.max(np.abs(XI)))
# Spatial window
if space_window:
x_center = (x[0] + x[-1]) / 2
L = (x[-1] - x[0]) / 2
window = np.exp(-((X - x_center)/L)**2)
sigma_vals *= window
exp_matrix = np.exp(1j * np.outer(x, xi)) # (Nx, Nxi)
integrand = sigma_vals * u_hat[np.newaxis, :] * exp_matrix
result = dxi * np.sum(integrand, axis=1) / (2 * np.pi)
return result
elif u_vals.ndim == 2:
# === 2D case ===
x1, x2 = x_grid
xi1, xi2 = xi_grid
dx1 = x1[1] - x1[0]
dx2 = x2[1] - x2[0]
dxi1 = xi1[1] - xi1[0]
dxi2 = xi2[1] - xi2[0]
X1, X2 = np.meshgrid(x1, x2, indexing='ij')
XI1, XI2 = np.meshgrid(xi1, xi2, indexing='ij')
# Fourier transform of u(x1, x2)
phase_ft = np.exp(-1j * (np.tensordot(x1, xi1, axes=0)[:, None, :, None] +
np.tensordot(x2, xi2, axes=0)[None, :, None, :]))
u_hat = np.tensordot(u_vals, phase_ft, axes=([0,1], [0,1])) * dx1 * dx2
# Symbol evaluation
sigma_vals = symbol_func(X1[:, :, None, None], X2[:, :, None, None],
XI1[None, None, :, :], XI2[None, None, :, :])
# Clamp values
sigma_vals = np.clip(sigma_vals, -clamp, clamp)
# Frequency window
if freq_window == 'gaussian':
sigma_xi1 = 0.8 * np.max(np.abs(XI1))
sigma_xi2 = 0.8 * np.max(np.abs(XI2))
window = np.exp(-(XI1[None, None, :, :] / sigma_xi1)**4 -
(XI2[None, None, :, :] / sigma_xi2)**4)
sigma_vals *= window
elif freq_window == 'hann':
# Frequency window - Hanning
wx = 0.5 * (1 + np.cos(np.pi * XI1 / np.max(np.abs(XI1))))
wy = 0.5 * (1 + np.cos(np.pi * XI2 / np.max(np.abs(XI2))))
# Mask to zero outside max frequency
mask_x = (np.abs(XI1) < np.max(np.abs(XI1)))
mask_y = (np.abs(XI2) < np.max(np.abs(XI2)))
# Expand wx and wy to match sigma_vals shape: (64, 64, 64, 64)
sigma_vals *= wx[:, :, None, None] * wy[:, :, None, None]
sigma_vals *= mask_x[:, :, None, None] * mask_y[:, :, None, None]
# Spatial window
if space_window:
x_center = (x1[0] + x1[-1])/2
y_center = (x2[0] + x2[-1])/2
Lx = (x1[-1] - x1[0])/2
Ly = (x2[-1] - x2[0])/2
window = np.exp(-((X1 - x_center)/Lx)**2 - ((X2 - y_center)/Ly)**2)
sigma_vals *= window[:, :, None, None]
# Oscillatory phase
phase = np.exp(1j * (X1[:, :, None, None] * XI1[None, None, :, :] +
X2[:, :, None, None] * XI2[None, None, :, :]))
integrand = sigma_vals * u_hat[None, None, :, :] * phase
result = dxi1 * dxi2 * np.sum(integrand, axis=(2, 3)) / (2 * np.pi)**2
return result
else:
raise NotImplementedError("Only 1D and 2D supported")
def step_ETD_RK4(self, u):
"""
Perform one Exponential Time Differencing Runge-Kutta of 4th order (ETD-RK4) time step
for first-order in time PDEs of the form:
∂ₜu = L u + N(u)
where L is a linear operator (possibly nonlocal or pseudo-differential), and N is a
nonlinear term treated via pseudo-spectral methods. This method evaluates the
exponential integrator up to fourth-order accuracy in time.
The ETD-RK4 scheme uses four stages to approximate the integral of the variation-of-constants formula:
uⁿ⁺¹ = e^(L Δt) uⁿ + Δt ∫₀¹ e^(L Δt (1 - τ)) φ(N(u(τ))) dτ
where φ denotes the nonlinear contributions evaluated at intermediate stages.
Parameters
u (np.ndarray): Current solution in real space (physical grid values).
Returns:
np.ndarray: Updated solution in real space after one ETD-RK4 time step.
Notes:
- The linear part L is diagonal in Fourier space and precomputed as self.L(k).
- Nonlinear terms are evaluated in physical space and transformed via FFT.
- The functions φ₁(z) and φ₂(z) are entire functions arising from the ETD scheme:
φ₁(z) = (eᶻ - 1)/z if z ≠ 0
= 1 if z = 0
φ₂(z) = (eᶻ - 1 - z)/z² if z ≠ 0
= ½ if z = 0
- This implementation assumes periodic boundary conditions and uses spectral differentiation via FFT.
- See Hochbruck & Ostermann (2010) for theoretical background on exponential integrators.
See Also:
step_ETD_RK4_order2 : For second-order in time equations.
psiOp_apply : For applying pseudo-differential operators.
apply_nonlinear : For handling nonlinear terms in the PDE.
"""
dt = self.dt
L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY)
E = np.exp(dt * L_fft)
E2 = np.exp(dt * L_fft / 2)
def phi1(z):
return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1) / z, 1.0)
def phi2(z):
return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1 - z) / z**2, 0.5)
phi1_dtL = phi1(dt * L_fft)
phi2_dtL = phi2(dt * L_fft)
fft = self.fft
ifft = self.ifft
u_hat = fft(u)
N1 = fft(self.apply_nonlinear(u))
a = ifft(E2 * (u_hat + 0.5 * dt * N1 * phi1_dtL))
N2 = fft(self.apply_nonlinear(a))
b = ifft(E2 * (u_hat + 0.5 * dt * N2 * phi1_dtL))
N3 = fft(self.apply_nonlinear(b))
c = ifft(E * (u_hat + dt * N3 * phi1_dtL))
N4 = fft(self.apply_nonlinear(c))
u_new_hat = E * u_hat + dt * (
N1 * phi1_dtL + 2 * (N2 + N3) * phi2_dtL + N4 * phi1_dtL
) / 6
return ifft(u_new_hat)
def step_ETD_RK4_order2(self, u, v):
"""
Perform one time step of the Exponential Time Differencing Runge-Kutta 4th-order (ETD-RK4) scheme for second-order PDEs.
This method evolves the solution u and its time derivative v forward in time by one step using the ETD-RK4 integrator.
It is designed for systems of the form:
∂ₜ²u = L u + N(u)
where L is a linear operator and N is a nonlinear term computed via self.apply_nonlinear.
The exponential integrator handles the linear part exactly in Fourier space, while the nonlinear terms are integrated
using a fourth-order Runge-Kutta-like approach. This ensures high accuracy and stability for stiff systems.
Parameters:
u (np.ndarray): Current solution array in real space.
v (np.ndarray): Current time derivative of the solution (∂ₜu) in real space.
Returns:
tuple: (u_new, v_new), updated solution and its time derivative after one time step.
Notes:
- Assumes periodic boundary conditions and uses FFT-based spectral methods.
- Handles both 1D and 2D problems seamlessly.
- Uses phi functions to compute exponential integrators efficiently.
- Suitable for wave equations and other second-order evolution equations with stiffness.
"""
dt = self.dt
L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY)
fft = self.fft
ifft = self.ifft
def rhs(u_val):
return ifft(L_fft * fft(u_val)) + self.apply_nonlinear(u_val, is_v=False)
# Stage A
A = rhs(u)
ua = u + 0.5 * dt * v
va = v + 0.5 * dt * A
# Stage B
B = rhs(ua)
ub = u + 0.5 * dt * va
vb = v + 0.5 * dt * B
# Stage C
C = rhs(ub)
uc = u + dt * vb
# Stage D
D = rhs(uc)
# Final update
u_new = u + dt * v + (dt**2 / 6.0) * (A + 2*B + 2*C + D)
v_new = v + (dt / 6.0) * (A + 2*B + 2*C + D)
return u_new, v_new
def check_cfl_condition(self):
"""
Check the CFL (Courant–Friedrichs–Lewymann) condition based on group velocity
for second-order time-dependent PDEs.
This method verifies whether the chosen time step dt satisfies the numerical stability
condition derived from the maximum wave propagation speed in the system. It supports both
1D and 2D problems, with or without a symbolic dispersion relation ω(k).
The CFL condition ensures that information does not propagate further than one grid cell
per time step. A safety factor of 0.5 is applied by default to ensure robustness.
Notes:
- In 1D, the group velocity v₉(k) = dω/dk is used to compute the maximum wave speed.
- In 2D, the x- and y-directional group velocities are evaluated independently.
- If no dispersion relation is available, the imaginary part of the linear operator L(k)
is used as an approximation for wave speed.
Raises:
-------
NotImplementedError:
If the spatial dimension is not 1D or 2D.
Prints:
-------
Warning message if the current time step dt exceeds the CFL-stable limit.
"""
print("\n*****************")
print("* CFL condition *")
print("*****************\n")
cfl_factor = 0.5 # Safety factor
if self.dim == 1:
if self.temporal_order == 2 and hasattr(self, 'omega'):
k_vals = self.kx
omega_vals = np.real(self.omega(k_vals))
with np.errstate(divide='ignore', invalid='ignore'):
v_group = np.gradient(omega_vals, k_vals)
max_speed = np.max(np.abs(v_group))
else:
max_speed = np.max(np.abs(np.imag(self.L(self.kx))))
dx = self.Lx / self.Nx
cfl_limit = cfl_factor * dx / max_speed if max_speed != 0 else np.inf
if self.dt > cfl_limit:
print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}")
elif self.dim == 2:
if self.temporal_order == 2 and hasattr(self, 'omega'):
k_vals = self.kx
omega_x = np.real(self.omega(k_vals, 0))
omega_y = np.real(self.omega(0, k_vals))
with np.errstate(divide='ignore', invalid='ignore'):
v_group_x = np.gradient(omega_x, k_vals)
v_group_y = np.gradient(omega_y, k_vals)
max_speed_x = np.max(np.abs(v_group_x))
max_speed_y = np.max(np.abs(v_group_y))
else:
max_speed_x = np.max(np.abs(np.imag(self.L(self.kx, 0))))
max_speed_y = np.max(np.abs(np.imag(self.L(0, self.ky))))
dx = self.Lx / self.Nx
dy = self.Ly / self.Ny
cfl_limit = cfl_factor / (max_speed_x / dx + max_speed_y / dy) if (max_speed_x + max_speed_y) != 0 else np.inf
if self.dt > cfl_limit:
print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}")
else:
raise NotImplementedError("Only 1D and 2D problems are supported.")
def check_symbol_conditions(self, k_range=None, verbose=True):
"""
Check strict analytic conditions on the linear symbol self.L_symbolic:
This method evaluates three key properties of the Fourier multiplier
symbol a(k) = self.L(k), which are crucial for well-posedness, stability,
and numerical efficiency. The checks apply to both 1D and 2D cases.
Conditions checked:
------------------
1. **Stability condition**: Re(a(k)) ≤ 0 for all k ≠ 0
Ensures that the system does not exhibit exponential growth in time.
2. **Dissipation condition**: Re(a(k)) ≤ -δ |k|² for large |k|
Ensures sufficient damping at high frequencies to avoid oscillatory instability.
3. **Growth condition**: |a(k)| ≤ C (1 + |k|)^m with m ≤ 4
Ensures that the symbol does not grow too rapidly with frequency,
which would otherwise cause numerical instability or unphysical amplification.
Parameters
----------
k_range : tuple or None, optional
Specifies the range of frequencies to test in the form (k_min, k_max, N).
If None, defaults are used: [-10, 10] with 500 points in 1D, or [-10, 10]
with 100 points per axis in 2D.
verbose : bool, default=True
If True, prints detailed results of each condition check.
Returns:
--------
None
Output is printed directly to the console for interpretability.
Notes:
------
- In 2D, the radial frequency |k| = √(kx² + ky²) is used for comparisons.
- The dissipation threshold assumes δ = 0.01 and p = 2 by default.
- The growth ratio is compared against |k|⁴; values above 100 indicate rapid growth.
- This function is typically called during solver setup or analysis phase.
See Also:
---------
analyze_wave_propagation : For further symbolic and numerical analysis of dispersion.
plot_symbol : Visualizes the symbol's behavior over the frequency domain.
"""
print("\n********************")
print("* Symbol condition *")
print("********************\n")
if self.dim == 1:
if k_range is None:
k_vals = np.linspace(-10, 10, 500)
else:
k_min, k_max, N = k_range
k_vals = np.linspace(k_min, k_max, N)
L_vals = self.L(k_vals)
k_abs = np.abs(k_vals)
elif self.dim == 2:
if k_range is None:
k_vals = np.linspace(-10, 10, 100)
else:
k_min, k_max, N = k_range
k_vals = np.linspace(k_min, k_max, N)
KX, KY = np.meshgrid(k_vals, k_vals)
L_vals = self.L(KX, KY)
k_abs = np.sqrt(KX**2 + KY**2)
else:
raise ValueError("Only 1D and 2D dimensions are supported.")
re_vals = np.real(L_vals)
abs_vals = np.abs(L_vals)
# === Condition 1: Stability
if np.any(re_vals > 1e-12):
max_pos = np.max(re_vals)
if verbose:
print(f"❌ Stability violated: max Re(a(k)) = {max_pos}")
print("Unstable symbol: Re(a(k)) > 0")
elif verbose:
print("✅ Spectral stability satisfied: Re(a(k)) ≤ 0")
# === Condition 2: Dissipation
mask = k_abs > 2
if np.any(mask):
re_decay = re_vals[mask]
expected_decay = -0.01 * k_abs[mask]**2
if np.any(re_decay > expected_decay + 1e-6):
if verbose:
print("⚠️ Insufficient high-frequency dissipation")
else:
if verbose:
print("✅ Proper high-frequency dissipation")
# === Condition 3: Growth
growth_ratio = abs_vals / (1 + k_abs)**4
if np.max(growth_ratio) > 100:
if verbose:
print("⚠️ Symbol grows rapidly: |a(k)| ≳ |k|^4")
else:
if verbose:
print("✅ Reasonable spectral growth")
if verbose:
print("✔ Symbol analysis completed.")
def analyze_wave_propagation(self):
"""
Perform a detailed analysis of wave propagation characteristics based on the dispersion relation ω(k).
This method visualizes key wave properties in both 1D and 2D settings:
- Dispersion relation: ω(k)
- Phase velocity: v_p(k) = ω(k)/|k|
- Group velocity: v_g(k) = ∇ₖ ω(k)
- Anisotropy in 2D (via magnitude of group velocity)
The symbolic dispersion relation 'omega_symbolic' must be defined beforehand.
This is typically available only for second-order-in-time equations.
In 1D:
Plots ω(k), v_p(k), and v_g(k) over a range of k values.
In 2D:
Displays heatmaps of ω(kx, ky), v_p(kx, ky), and |v_g(kx, ky)| over a 2D wavenumber grid.
Raises:
AttributeError: If 'omega_symbolic' is not defined, the method exits gracefully with a message.
Side Effects:
Generates and displays matplotlib plots.
"""
print("\n*****************************")
print("* Wave propagation analysis *")
print("*****************************\n")
if not hasattr(self, 'omega_symbolic'):
print("❌ omega_symbolic not defined. Only available for 2nd order in time.")
return
if self.dim == 1:
k = self.k_symbols[0]
omega_func = lambdify(k, self.omega_symbolic, 'numpy')
k_vals = np.linspace(-10, 10, 1000)
omega_vals = omega_func(k_vals)
with np.errstate(divide='ignore', invalid='ignore'):
v_phase = np.where(k_vals != 0, omega_vals / k_vals, 0.0)
dk = k_vals[1] - k_vals[0]
v_group = np.gradient(omega_vals, dk)
plt.figure(figsize=(10, 6))
plt.plot(k_vals, omega_vals, label=r'$\omega(k)$')
plt.plot(k_vals, v_phase, label=r'$v_p(k)$')
plt.plot(k_vals, v_group, label=r'$v_g(k)$')
plt.title("1D Wave Propagation Analysis")
plt.xlabel("k")
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()
elif self.dim == 2:
kx, ky = self.k_symbols
omega_func = lambdify((kx, ky), self.omega_symbolic, 'numpy')
k_vals = np.linspace(-10, 10, 200)
KX, KY = np.meshgrid(k_vals, k_vals)
K_mag = np.sqrt(KX**2 + KY**2)
K_mag[K_mag == 0] = 1e-8 # Avoid division by 0
omega_vals = omega_func(KX, KY)
v_phase = np.real(omega_vals) / K_mag
dk = k_vals[1] - k_vals[0]
domega_dx = np.gradient(omega_vals, dk, axis=0)
domega_dy = np.gradient(omega_vals, dk, axis=1)
v_group_norm = np.sqrt(np.abs(domega_dx)**2 + np.abs(domega_dy)**2)
fig, axs = plt.subplots(1, 3, figsize=(18, 5))
im0 = axs[0].imshow(np.real(omega_vals), extent=[-10, 10, -10, 10],
origin='lower', cmap='viridis')
axs[0].set_title(r'$\omega(k_x, k_y)$')
plt.colorbar(im0, ax=axs[0])
im1 = axs[1].imshow(v_phase, extent=[-10, 10, -10, 10],
origin='lower', cmap='plasma')
axs[1].set_title(r'$v_p(k_x, k_y)$')
plt.colorbar(im1, ax=axs[1])
im2 = axs[2].imshow(v_group_norm, extent=[-10, 10, -10, 10],
origin='lower', cmap='inferno')
axs[2].set_title(r'$|v_g(k_x, k_y)|$')
plt.colorbar(im2, ax=axs[2])
for ax in axs:
ax.set_xlabel(r'$k_x$')
ax.set_ylabel(r'$k_y$')
ax.set_aspect('equal')
plt.tight_layout()
plt.show()
else:
print("❌ Only 1D and 2D wave analysis supported.")
def plot_symbol(self, component="abs", k_range=None, cmap="viridis"):
"""
Visualize the spectral symbol L(k) or L(kx, ky) in 1D or 2D.
This method plots the linear operator's symbolic Fourier representation
either as a function of a single wavenumber k (1D), or two wavenumbers
kx and ky (2D). The user can choose to display the real part, imaginary part,
or absolute value of the symbol.
Parameters
----------
component : str {'abs', 're', 'im'}
Component of the symbol to visualize:
- 'abs' : absolute value |a(k)|
- 're' : real part Re[a(k)]
- 'im' : imaginary part Im[a(k)]
k_range : tuple (kmin, kmax, N), optional
Wavenumber range for evaluation:
- kmin: minimum wavenumber
- kmax: maximum wavenumber
- N: number of sampling points
If None, defaults to [-10, 10] with high resolution.
cmap : str, optional
Colormap used for 2D surface plots. Default is 'viridis'.
Raises
------
ValueError: If the spatial dimension is not 1D or 2D.
Notes:
- In 1D, the symbol is plotted using a standard 2D line plot.
- In 2D, a 3D surface plot is generated with color-mapped height.
- Symbol evaluation uses self.L(k), which must be defined and callable.
"""
print("\n*******************")
print("* Symbol plotting *")
print("*******************\n")
assert component in ("abs", "re", "im"), "component must be 'abs', 're' or 'im'"
if self.dim == 1:
if k_range is None:
k_vals = np.linspace(-10, 10, 1000)
else:
kmin, kmax, N = k_range
k_vals = np.linspace(kmin, kmax, N)
L_vals = self.L(k_vals)
if component == "re":
vals = np.real(L_vals)
label = "Re[a(k)]"
elif component == "im":
vals = np.imag(L_vals)
label = "Im[a(k)]"
else:
vals = np.abs(L_vals)
label = "|a(k)|"
plt.plot(k_vals, vals)
plt.xlabel("k")
plt.ylabel(label)
plt.title(f"Spectral symbol: {label}")
plt.grid(True)
plt.show()
elif self.dim == 2:
if k_range is None:
k_vals = np.linspace(-10, 10, 300)
else:
kmin, kmax, N = k_range
k_vals = np.linspace(kmin, kmax, N)
KX, KY = np.meshgrid(k_vals, k_vals)
L_vals = self.L(KX, KY)
if component == "re":
Z = np.real(L_vals)
title = "Re[a(kx, ky)]"
elif component == "im":
Z = np.imag(L_vals)
title = "Im[a(kx, ky)]"
else:
Z = np.abs(L_vals)
title = "|a(kx, ky)|"
fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
surf = ax.plot_surface(KX, KY, Z, cmap=cmap, edgecolor='none', antialiased=True)
fig.colorbar(surf, ax=ax, shrink=0.6)
ax.set_xlabel("kx")
ax.set_ylabel("ky")
ax.set_zlabel(title)
ax.set_title(f"2D spectral symbol: {title}")
plt.tight_layout()
plt.show()
else:
raise ValueError("Only 1D and 2D supported.")
def compute_energy(self):
"""
Compute the total energy of the wave equation solution for second-order temporal PDEs.
The energy is defined as:
E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹ᐟ²u|² ] dx
where L is the linear operator associated with the spatial part of the PDE,
and L¹ᐟ² denotes its square root in Fourier space.
This method supports both 1D and 2D problems and is only meaningful when
self.temporal_order == 2 (second-order time derivative).
Returns
-------
float or None:
Total energy at current time step. Returns None if the temporal order is not 2 or if no valid velocity data (v_prev) is available.
Notes
-----
- Uses FFT-based spectral differentiation to compute the spatial contributions.
- Assumes periodic boundary conditions.
- Handles both real and complex-valued solutions.
"""
if self.temporal_order != 2 or self.v_prev is None:
return None
u = self.u_prev
v = self.v_prev
# Fourier transform of u
u_hat = self.fft(u)
if self.dim == 1:
# 1D case
L_vals = self.L(self.KX)
sqrt_L = np.sqrt(np.abs(L_vals))
Lu_hat = sqrt_L * u_hat # Apply sqrt(|L(k)|) in Fourier space
Lu = self.ifft(Lu_hat)
dx = self.Lx / self.Nx
energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2)
total_energy = np.sum(energy_density) * dx
elif self.dim == 2:
# 2D case
L_vals = self.L(self.KX, self.KY)
sqrt_L = np.sqrt(np.abs(L_vals))
Lu_hat = sqrt_L * u_hat
Lu = self.ifft(Lu_hat)
dx = self.Lx / self.Nx
dy = self.Ly / self.Ny
energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2)
total_energy = np.sum(energy_density) * dx * dy
else:
raise ValueError("Unsupported dimension for u.")
return total_energy
def plot_energy(self, log=False):
"""
Plot the time evolution of the total energy for wave equations.
Visualizes the energy computed during simulation for both 1D and 2D cases.
Requires temporal_order=2 and prior execution of compute_energy() during solve().
Parameters:
log : bool
If True, displays energy on a logarithmic scale to highlight exponential decay/growth.
Notes:
- Energy is defined as E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹⸍²u|² ] dx
- Only available if energy monitoring was activated in solve()
- Automatically skips plotting if no energy data is available
Displays:
- Time vs. Total Energy plot with grid and legend
- Appropriate axis labels and dimensional context (1D/2D)
- Logarithmic or linear scaling based on input parameter
"""
if not hasattr(self, 'energy_history') or not self.energy_history:
print("No energy data recorded. Call compute_energy() within solve().")
return
# Time vector for plotting
t = np.linspace(0, self.Lt, len(self.energy_history))
# Create the figure
plt.figure(figsize=(6, 4))
if log:
plt.semilogy(t, self.energy_history, label="Energy (log scale)")
else:
plt.plot(t, self.energy_history, label="Energy")
# Axis labels and title
plt.xlabel("Time")
plt.ylabel("Total energy")
plt.title("Energy evolution ({}D)".format(self.dim))
# Display options
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()
def show_stationary_solution(self, u=None, component='abs', cmap='viridis'):
"""
Display the stationary solution computed by solve_stationary_psiOp.
This method visualizes the solution of a pseudo-differential equation
solved in stationary mode. It supports both 1D and 2D spatial domains,
with options to display different components of the solution (real,
imaginary, absolute value, or phase).
Parameters
----------
u : ndarray, optional
Precomputed solution array. If None, calls solve_stationary_psiOp()
to compute the solution.
component : str, optional {'real', 'imag', 'abs', 'angle'}
Component of the complex-valued solution to display:
- 'real': Real part
- 'imag': Imaginary part
- 'abs' : Absolute value (modulus)
- 'angle' : Phase (argument)
cmap : str, optional
Colormap used for 2D visualization (default: 'viridis').
Raises
------
ValueError
If an invalid component is specified or if the spatial dimension
is not supported (only 1D and 2D are implemented).
Notes
-----
- In 1D, the solution is displayed using a standard line plot.
- In 2D, the solution is visualized as a 3D surface plot.
"""
def get_component(u):
if component == 'real':
return np.real(u)
elif component == 'imag':
return np.imag(u)
elif component == 'abs':
return np.abs(u)
elif component == 'angle':
return np.angle(u)
else:
raise ValueError("Invalid component")
if u is None:
u = self.solve_stationary_psiOp()
if self.dim == 1:
# Plot the solution in 1D
plt.figure(figsize=(8, 4))
plt.plot(self.x_grid, get_component(u), label=f'{component} of u')
plt.xlabel('x')
plt.ylabel(f'{component} of u')
plt.title('Stationary solution (1D)')
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()
elif self.dim == 2:
fig = plt.figure(figsize=(12, 6))
ax = fig.add_subplot(111, projection='3d')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel(f'{component.title()} of u')
plt.title('Stationary solution (2D)')
data0 = get_component(u)
ax.plot_surface(self.X, self.Y, data0, cmap='viridis')
plt.tight_layout()
plt.show()
else:
raise ValueError("Only 1D and 2D display are supported.")
def animate(self, component='abs', overlay='contour'):
"""
Create an animated plot of the solution evolution over time.
This method generates a dynamic visualization of the solution array `self.frames`,
animating either the real part, imaginary part, absolute value, or complex angle
of the field. It supports both 1D line plots and 2D surface plots with optional
contour overlays.
Parameters
----------
component : str in {'real', 'imag', 'abs', 'angle'}
The component of the solution to visualize:
- 'real' : Real part Re(u)
- 'imag' : Imaginary part Im(u)
- 'abs' : Absolute value |u|
- 'angle' : Complex argument arg(u)
overlay : str in {'contour', 'front'}, optional
Type of overlay for 2D animations:
- 'contour' : Adds contour lines beneath the surface at each frame.
- 'front' : Used for tracking wavefronts.
Returns
-------
FuncAnimation
A Matplotlib `FuncAnimation` object that can be displayed or saved as a video.
Notes
-----
- Uses linear interpolation to map simulation frames to target animation frames.
- In 2D, the z-axis dynamically rescales based on current data range.
- For 'angle' component, color scaling is fixed between -π and π for consistency.
- The animation interval is fixed at 50 ms per frame for smooth playback.
"""
def get_component(u):
if component == 'real':
return np.real(u)
elif component == 'imag':
return np.imag(u)
elif component == 'abs':
return np.abs(u)
elif component == 'angle':
return np.angle(u)
else:
raise ValueError("Invalid component")
print("\n*********************")
print("* Solution plotting *")
print("*********************\n")
# === Calculate time vector of stored frames ===
save_interval = max(1, self.Nt // self.n_frames)
frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt)
# === Target times for animation ===
target_times = np.linspace(0, self.Lt, self.n_frames // 2)
# Map target times to nearest frame indices
frame_indices = [np.argmin(np.abs(frame_times - t)) for t in target_times]
if self.dim == 1:
fig, ax = plt.subplots()
line, = ax.plot(self.X, get_component(self.frames[0]))
ax.set_ylim(np.min(self.frames[0]), np.max(self.frames[0]))
ax.set_xlabel('x')
ax.set_ylabel(f'{component} of u')
ax.set_title('Initial condition')
plt.tight_layout()
plt.show()
def update(frame_number):
frame = frame_indices[frame_number]
ydata = get_component(self.frames[frame])
ydata_real = np.real(ydata) if np.iscomplexobj(ydata) else ydata
line.set_ydata(ydata_real)
ax.set_ylim(np.min(ydata_real), np.max(ydata_real))
current_time = target_times[frame_number]
ax.set_title(f't = {current_time:.2f}')
return line,
ani = FuncAnimation(fig, update, frames=len(target_times), interval=50)
return ani
else: # dim == 2
fig = plt.figure(figsize=(14, 8))
ax = fig.add_subplot(111, projection='3d')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel(f'{component.title()} of u')
ax.zaxis.labelpad = 0
ax.set_title('Initial condition')
data0 = get_component(self.frames[0])
surf = [ax.plot_surface(self.X, self.Y, data0, cmap='viridis')]
plt.show()
def update(frame_number):
frame = frame_indices[frame_number]
current_data = get_component(self.frames[frame])
z_offset = np.max(current_data) + 0.05 * (np.max(current_data) - np.min(current_data))
ax.clear()
surf[0] = ax.plot_surface(self.X, self.Y, current_data,
cmap='viridis', vmin=-1, vmax=1 if component != 'angle' else np.pi)
if overlay == 'contour':
ax.contour(self.X, self.Y, current_data, levels=10, cmap='cool', offset=z_offset)
elif overlay == 'front':
dx = self.x_grid[1] - self.x_grid[0]
dy = self.y_grid[1] - self.y_grid[0]
du_dx, du_dy = np.gradient(current_data, dx, dy)
grad_norm = np.sqrt(du_dx**2 + du_dy**2)
local_max = (grad_norm == maximum_filter(grad_norm, size=5))
normalized = grad_norm[local_max] / np.max(grad_norm)
colors = cm.plasma(normalized)
ax.scatter(self.X[local_max], self.Y[local_max],
z_offset * np.ones_like(self.X[local_max]),
color=colors, s=10, alpha=0.8)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel(f'{component.title()} of u')
current_time = target_times[frame_number]
ax.set_title(f'Solution at t = {current_time:.2f}')
return surf
ani = FuncAnimation(fig, update, frames=len(target_times), interval=50)
return ani
def test(self, u_exact, t_eval=None, norm='relative', threshold=1e-2, plot=True, component='real'):
"""
Test the solver against an exact solution.
This method quantitatively compares the numerical solution with a provided exact solution
at a specified time using either relative or absolute error norms. It supports both
stationary and time-dependent problems in 1D and 2D. If enabled, it also generates plots
of the solution, exact solution, and pointwise error.
Parameters
----------
u_exact : callable
Exact solution function taking spatial coordinates and optionally time as arguments.
t_eval : float, optional
Time at which to compare solutions. For non-stationary problems, defaults to final time Lt.
Ignored for stationary problems.
norm : str {'relative', 'absolute'}
Type of error norm used in comparison.
threshold : float
Acceptable error threshold; raises an assertion if exceeded.
plot : bool
Whether to display visual comparison plots (default: True).
component : str {'real', 'imag', 'abs'}
Component of the solution to compare and visualize.
Raises
------
ValueError
If unsupported dimension is encountered or requested evaluation time exceeds simulation duration.
AssertionError
If computed error exceeds the given threshold.
Prints
------
- Information about the closest available frame to the requested evaluation time.
- Computed error value and comparison to threshold.
Notes
-----
- For time-dependent problems, the solution is extracted from precomputed frames.
- Plots are adapted to spatial dimension: line plots for 1D, image plots for 2D.
- The method ensures consistent handling of real, imaginary, and magnitude components.
"""
if self.is_stationary:
print("Testing a stationary solution.")
u_num = self.u
# Compute exact solution
if self.dim == 1:
u_ex = u_exact(self.X)
elif self.dim == 2:
u_ex = u_exact(self.X, self.Y)
else:
raise ValueError("Unsupported dimension.")
actual_t = None
else:
if t_eval is None:
t_eval = self.Lt
save_interval = max(1, self.Nt // self.n_frames)
frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt)
frame_index = np.argmin(np.abs(frame_times - t_eval))
actual_t = frame_times[frame_index]
print(f"Closest available time to t_eval={t_eval}: {actual_t}")
if frame_index >= len(self.frames):
raise ValueError(f"Time t = {t_eval} exceeds simulation duration.")
u_num = self.frames[frame_index]
# Compute exact solution at the actual time
if self.dim == 1:
u_ex = u_exact(self.X, actual_t)
elif self.dim == 2:
u_ex = u_exact(self.X, self.Y, actual_t)
else:
raise ValueError("Unsupported dimension.")
# Select component
if component == 'real':
diff = np.real(u_num) - np.real(u_ex)
ref = np.real(u_ex)
elif component == 'imag':
diff = np.imag(u_num) - np.imag(u_ex)
ref = np.imag(u_ex)
elif component == 'abs':
diff = np.abs(u_num) - np.abs(u_ex)
ref = np.abs(u_ex)
else:
raise ValueError("Invalid component.")
# Compute error
if norm == 'relative':
error = np.linalg.norm(diff) / np.linalg.norm(ref)
elif norm == 'absolute':
error = np.linalg.norm(diff)
else:
raise ValueError("Unknown norm type.")
label_time = f"t = {actual_t}" if actual_t is not None else ""
print(f"Test error {label_time}: {error:.3e}")
assert error < threshold, f"Error too large {label_time}: {error:.3e}"
# Plot
if plot:
if self.dim == 1:
plt.figure(figsize=(12, 6))
plt.subplot(2, 1, 1)
plt.plot(self.X, np.real(u_num), label='Numerical')
plt.plot(self.X, np.real(u_ex), '--', label='Exact')
plt.title(f'Solution {label_time}, error = {error:.2e}')
plt.legend()
plt.grid()
plt.subplot(2, 1, 2)
plt.plot(self.X, np.abs(diff), color='red')
plt.title('Absolute Error')
plt.grid()
plt.tight_layout()
plt.show()
else:
plt.figure(figsize=(15, 5))
plt.subplot(1, 3, 1)
plt.title("Numerical Solution")
plt.imshow(np.abs(u_num), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis')
plt.colorbar()
plt.subplot(1, 3, 2)
plt.title("Exact Solution")
plt.imshow(np.abs(u_ex), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis')
plt.colorbar()
plt.subplot(1, 3, 3)
plt.title(f"Error (Norm = {error:.2e})")
plt.imshow(np.abs(diff), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='inferno')
plt.colorbar()
plt.tight_layout()
plt.show()Classes
class Op (*args)-
Custom symbolic wrapper for pseudo-differential operators in Fourier space. Usage: Op(symbol_expr, u)
Expand source code
class Op(Function): """Custom symbolic wrapper for pseudo-differential operators in Fourier space. Usage: Op(symbol_expr, u) """ nargs = 2Ancestors
- sympy.core.function.Function
- sympy.core.function.Application
- sympy.core.expr.Expr
- sympy.core.basic.Basic
- sympy.printing.defaults.Printable
- sympy.core.evalf.EvalfMixin
Class variables
var default_assumptionsvar nargs
class PDESolver (equation, time_scheme='default', dealiasing_ratio=0.6666666666666666)-
A partial differential equation (PDE) solver based on spectral methods using Fourier transforms.
This solver supports symbolic specification of PDEs via SymPy and numerical solution using high-order spectral techniques. It is designed for both linear and nonlinear time-dependent PDEs, as well as stationary pseudo-differential problems.
Key Features:
- Symbolic PDE parsing using SymPy expressions
- 1D and 2D spatial domains with periodic boundary conditions
- Fourier-based spectral discretization with dealiasing
- Temporal integration schemes:
- Default exponential time stepping
- ETD-RK4 (Exponential Time Differencing Runge-Kutta of 4th order)
- Nonlinear terms handled through pseudo-spectral evaluation
- Built-in tools for:
- Visualization of solutions and error surfaces
- Symbol analysis of linear and pseudo-differential operators
- Microlocal analysis (e.g., wavefront set estimation, Hamiltonian flows)
- CFL condition checking and numerical stability diagnostics
Supported Operators:
- Linear differential and pseudo-differential operators
- Nonlinear terms up to second order in derivatives
- Symbolic operator composition and adjoints
- Asymptotic inversion of elliptic operators for stationary problems
Example Usage:
>>> from PDESolver import * >>> u = Function('u') >>> t, x = symbols('t x') >>> eq = Eq(diff(u(t, x), t), diff(u(t, x), x, 2) + u(t, x)**2) >>> def initial(x): return np.sin(x) >>> solver = PDESolver(eq) >>> solver.setup(Lx=2*np.pi, Nx=128, Lt=1.0, Nt=1000, initial_condition=initial) >>> solver.solve() >>> ani = solver.animate() >>> HTML(ani.to_jshtml()) # Display animation in Jupyter notebookInitialize the PDE solver with a given equation.
This method analyzes the input partial differential equation (PDE), identifies the unknown function and its dependencies, determines whether the problem is stationary or time-dependent, and prepares symbolic and numerical structures for solving in spectral space.
Supported features:
- 1D and 2D problems
- Time-dependent and stationary equations
- Linear and nonlinear terms
- Pseudo-differential operators via
psiOp - Source terms and boundary conditions
The equation is parsed to extract linear, nonlinear, source, and pseudo-differential components. Symbolic manipulation is used to derive the Fourier representation of linear operators when applicable.
Parameters
equation:sympy.Eq- The PDE expressed as a SymPy equation.
time_scheme:str- Temporal integration scheme: - 'default' for exponential - time-stepping or 'ETD-RK4' for fourth-order exponential - time differencing Runge–Kutta.
dealiasing_ratio:float- Fraction of high-frequency modes to zero out during dealiasing (e.g., 2/3 for standard truncation).
Attributes initialized:
- self.u: the unknown function (e.g., u(t, x))
- self.dim: spatial dimension (1 or 2)
- self.spatial_vars: list of spatial variables (e.g., [x] or [x, y])
- self.is_stationary: boolean indicating if the problem is stationary
- self.linear_terms: dictionary mapping derivative orders to coefficients
- self.nonlinear_terms: list of nonlinear expressions
- self.source_terms: list of source functions
- self.pseudo_terms: list of pseudo-differential operator expressions
- self.has_psi: boolean indicating presence of pseudo-differential operators
- self.fft / self.ifft: appropriate FFT routines based on spatial dimension
- self.kx, self.ky: symbolic wavenumber variables for Fourier space
Raises
ValueError- If the equation does not contain exactly one unknown function, if unsupported dimensions are detected, or invalid dependencies.
Expand source code
class PDESolver: """ A partial differential equation (PDE) solver based on **spectral methods** using Fourier transforms. This solver supports symbolic specification of PDEs via SymPy and numerical solution using high-order spectral techniques. It is designed for both **linear and nonlinear time-dependent PDEs**, as well as **stationary pseudo-differential problems**. Key Features: ------------- - Symbolic PDE parsing using SymPy expressions - 1D and 2D spatial domains with periodic boundary conditions - Fourier-based spectral discretization with dealiasing - Temporal integration schemes: - Default exponential time stepping - ETD-RK4 (Exponential Time Differencing Runge-Kutta of 4th order) - Nonlinear terms handled through pseudo-spectral evaluation - Built-in tools for: - Visualization of solutions and error surfaces - Symbol analysis of linear and pseudo-differential operators - Microlocal analysis (e.g., wavefront set estimation, Hamiltonian flows) - CFL condition checking and numerical stability diagnostics Supported Operators: -------------------- - Linear differential and pseudo-differential operators - Nonlinear terms up to second order in derivatives - Symbolic operator composition and adjoints - Asymptotic inversion of elliptic operators for stationary problems Example Usage: -------------- >>> from PDESolver import * >>> u = Function('u') >>> t, x = symbols('t x') >>> eq = Eq(diff(u(t, x), t), diff(u(t, x), x, 2) + u(t, x)**2) >>> def initial(x): return np.sin(x) >>> solver = PDESolver(eq) >>> solver.setup(Lx=2*np.pi, Nx=128, Lt=1.0, Nt=1000, initial_condition=initial) >>> solver.solve() >>> ani = solver.animate() >>> HTML(ani.to_jshtml()) # Display animation in Jupyter notebook """ def __init__(self, equation, time_scheme='default', dealiasing_ratio=2/3): """ Initialize the PDE solver with a given equation. This method analyzes the input partial differential equation (PDE), identifies the unknown function and its dependencies, determines whether the problem is stationary or time-dependent, and prepares symbolic and numerical structures for solving in spectral space. Supported features: - 1D and 2D problems - Time-dependent and stationary equations - Linear and nonlinear terms - Pseudo-differential operators via `psiOp` - Source terms and boundary conditions The equation is parsed to extract linear, nonlinear, source, and pseudo-differential components. Symbolic manipulation is used to derive the Fourier representation of linear operators when applicable. Parameters ---------- equation : sympy.Eq The PDE expressed as a SymPy equation. time_scheme : str Temporal integration scheme: - 'default' for exponential - time-stepping or 'ETD-RK4' for fourth-order exponential - time differencing Runge–Kutta. dealiasing_ratio : float Fraction of high-frequency modes to zero out during dealiasing (e.g., 2/3 for standard truncation). Attributes initialized: - self.u: the unknown function (e.g., u(t, x)) - self.dim: spatial dimension (1 or 2) - self.spatial_vars: list of spatial variables (e.g., [x] or [x, y]) - self.is_stationary: boolean indicating if the problem is stationary - self.linear_terms: dictionary mapping derivative orders to coefficients - self.nonlinear_terms: list of nonlinear expressions - self.source_terms: list of source functions - self.pseudo_terms: list of pseudo-differential operator expressions - self.has_psi: boolean indicating presence of pseudo-differential operators - self.fft / self.ifft: appropriate FFT routines based on spatial dimension - self.kx, self.ky: symbolic wavenumber variables for Fourier space Raises: ValueError: If the equation does not contain exactly one unknown function, if unsupported dimensions are detected, or invalid dependencies. """ self.time_scheme = time_scheme # 'default' or 'ETD-RK4' self.dealiasing_ratio = dealiasing_ratio print("\n*********************************") print("* Partial differential equation *") print("*********************************\n") pprint(equation) # Extract symbols and function from the equation functions = equation.atoms(Function) # Ignore the wrappers psiOp and Op excluded_wrappers = {'psiOp', 'Op'} # Extract the candidate fonctions (excluding wrappers) candidate_functions = [ f for f in functions if f.func.__name__ not in excluded_wrappers ] # Keep only user functions (u(x), u(x, t), etc.) candidate_functions = [ f for f in functions if isinstance(f, AppliedUndef) ] # Stationary detection: no dependence on t self.is_stationary = all( not any(str(arg) == 't' for arg in f.args) for f in candidate_functions ) if len(candidate_functions) != 1: print("candidate_functions :", candidate_functions) raise ValueError("The equation must contain exactly one unknown function") self.u = candidate_functions[0] self.u_eq = self.u args = self.u.args if self.is_stationary: if len(args) not in (1, 2): raise ValueError("Stationary problems must depend on 1 or 2 spatial variables") self.spatial_vars = args else: if len(args) < 2 or len(args) > 3: raise ValueError("The function must depend on t and at least one spatial variable (x [, y])") self.t = args[0] self.spatial_vars = args[1:] self.dim = len(self.spatial_vars) if self.dim == 1: self.x = self.spatial_vars[0] self.y = None elif self.dim == 2: self.x, self.y = self.spatial_vars else: raise ValueError("Only 1D and 2D problems are supported.") if self.dim == 1: self.fft = partial(fft, workers=FFT_WORKERS) self.ifft = partial(ifft, workers=FFT_WORKERS) else: self.fft = partial(fft2, workers=FFT_WORKERS) self.ifft = partial(ifft2, workers=FFT_WORKERS) # Parse the equation self.linear_terms = {} self.nonlinear_terms = [] self.symbol_terms = [] self.source_terms = [] self.pseudo_terms = [] self.temporal_order = 0 # Order of the temporal derivative self.linear_terms, self.nonlinear_terms, self.symbol_terms, self.source_terms, self.pseudo_terms = self.parse_equation(equation) # flag : pseudo‑differential operator present ? self.has_psi = bool(self.pseudo_terms) if self.has_psi: print('⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.') self.is_spatial = False for coeff, expr in self.pseudo_terms: if expr.has(self.x) or (self.dim == 2 and expr.has(self.y)): self.is_spatial = True break if self.dim == 1: self.kx = symbols('kx') elif self.dim == 2: self.kx, self.ky = symbols('kx ky') # Compute linear operator if not self.is_stationary: self.compute_linear_operator() else: self.psi_ops = [] for coeff, sym_expr in self.pseudo_terms: psi = PseudoDifferentialOperator(sym_expr, self.spatial_vars, self.u, mode='symbol') self.psi_ops.append((coeff, psi)) def parse_equation(self, equation): """ Parse the PDE to separate linear and nonlinear terms, symbolic operators (Op), source terms, and pseudo-differential operators (psiOp). This method rewrites the input equation in standard form (lhs - rhs = 0), expands it, and classifies each term into one of the following categories: - Linear terms involving derivatives or the unknown function u - Nonlinear terms (products with u, powers of u, etc.) - Symbolic pseudo-differential operators (Op) - Source terms (independent of u) - Pseudo-differential operators (psiOp) Parameters equation (sympy.Eq): The partial differential equation to be analyzed. Can be provided as an Eq object or a sympy expression. Returns: tuple: A 5-tuple containing: - linear_terms (dict): Mapping from derivative/function to coefficient. - nonlinear_terms (list): List of terms classified as nonlinear. - symbol_terms (list): List of (coefficient, symbolic operator) pairs. - source_terms (list): List of terms independent of the unknown function. - pseudo_terms (list): List of (coefficient, pseudo-differential symbol) pairs. Notes: - If `psiOp` is present in the equation, expansion is skipped for safety. - When `psiOp` is used, only nonlinear terms, source terms, and possibly a time derivative are allowed; other linear terms and symbolic operators (Op) are forbidden. - Classification logic includes: - Detection of nonlinear structures like products or powers of u - Mixed terms involving both u and its derivatives - External symbolic operators (Op) and pseudo-differential operators (psiOp) """ def is_nonlinear_term(term, u_func): # If the term contains functions (Abs, sin, exp, ...) applied to u if term.has(u_func): for sub in preorder_traversal(term): if isinstance(sub, Function) and sub.has(u_func) and sub.func != u_func.func: return True # If the term contains a nonlinear power of u if term.has(Pow): for pow_term in term.atoms(Pow): if pow_term.base == u_func and pow_term.exp != 1: return True # If the term is a product containing u and its derivative if term.func == Mul: factors = term.args has_u = any((f.has(u_func) and not isinstance(f, Derivative) for f in factors)) has_derivative = any((isinstance(f, Derivative) and f.expr.func == u_func.func for f in factors)) if has_u and has_derivative: return True return False print("\n********************") print("* Equation parsing *") print("********************\n") if isinstance(equation, Eq): lhs = equation.lhs - equation.rhs else: lhs = equation print(f"\nEquation rewritten in standard form: {lhs}") if lhs.has(psiOp): print("⚠️ psiOp detected: skipping expansion for safety") lhs_expanded = lhs else: lhs_expanded = expand(lhs) print(f"\nExpanded equation: {lhs_expanded}") linear_terms = {} nonlinear_terms = [] symbol_terms = [] source_terms = [] pseudo_terms = [] for term in lhs_expanded.as_ordered_terms(): print(f"Analyzing term: {term}") if isinstance(term, psiOp): expr = term.args[0] pseudo_terms.append((1, expr)) print(" --> Classified as pseudo linear term (psiOp)") continue # Otherwise, look for psiOp inside (general case) if term.has(psiOp): psiops = term.atoms(psiOp) for psi in psiops: try: coeff = simplify(term / psi) expr = psi.args[0] pseudo_terms.append((coeff, expr)) print(" --> Classified as pseudo linear term (psiOp)") except Exception as e: print(f" ⚠️ Failed to extract psiOp coefficient in term: {term}") print(f" Reason: {e}") nonlinear_terms.append(term) print(" --> Fallback: classified as nonlinear") continue if term.has(Op): ops = term.atoms(Op) for op in ops: coeff = term / op expr = op.args[0] symbol_terms.append((coeff, expr)) print(" --> Classified as symbolic linear term (Op)") continue if is_nonlinear_term(term, self.u): nonlinear_terms.append(term) print(" --> Classified as nonlinear") continue derivs = term.atoms(Derivative) if derivs: deriv = derivs.pop() coeff = term / deriv linear_terms[deriv] = linear_terms.get(deriv, 0) + coeff print(f" Derivative found: {deriv}") print(" --> Classified as linear") elif self.u in term.atoms(Function): coeff = term.as_coefficients_dict().get(self.u, 1) linear_terms[self.u] = linear_terms.get(self.u, 0) + coeff print(" --> Classified as linear") else: source_terms.append(term) print(" --> Classified as source term") print(f"Final linear terms: {linear_terms}") print(f"Final nonlinear terms: {nonlinear_terms}") print(f"Symbol terms: {symbol_terms}") print(f"Pseudo terms: {pseudo_terms}") print(f"Source terms: {source_terms}") if pseudo_terms: # Check if a time derivative is present among the linear terms has_time_derivative = any( isinstance(term, Derivative) and self.t in [v for v, _ in term.variable_count] for term in linear_terms ) # Extract non-temporal linear terms invalid_linear_terms = { term: coeff for term, coeff in linear_terms.items() if not ( isinstance(term, Derivative) and self.t in [v for v, _ in term.variable_count] ) and term != self.u # exclusion of the simple u term (without derivative) } if invalid_linear_terms or symbol_terms: raise ValueError( "When psiOp is used, only nonlinear terms, source terms, " "and possibly a time derivative are allowed. " "Other linear terms and Ops are forbidden." ) return linear_terms, nonlinear_terms, symbol_terms, source_terms, pseudo_terms def compute_linear_operator(self): """ Compute the symbolic Fourier representation L(k) of the linear operator derived from the linear part of the PDE. This method constructs a dispersion relation by applying each symbolic derivative to a plane wave exp(i(k·x - ωt)) and extracting the resulting expression. It handles arbitrary derivative combinations and includes symbolic and pseudo-differential terms. Steps: ------- 1. Construct a plane wave φ(x, t) = exp(i(k·x - ωt)). 2. Apply each term from self.linear_terms to φ. 3. Normalize by φ and simplify to obtain L(k). 4. Include symbolic terms (e.g., psiOp) if present. 5. Detect the temporal order from the dispersion relation. 6. Build the numerical function L(k) via lambdify. Sets: ----- - self.L_symbolic : sympy.Expr Symbolic form of L(k). - self.L : callable Numerical function of L(kx[, ky]). - self.omega : callable or None Frequency root ω(k), if available. - self.temporal_order : int Order of time derivatives detected. - self.psi_ops : list of (coeff, PseudoDifferentialOperator) Pseudo-differential terms present in the equation. Raises: ------- ValueError if the dimension is unsupported or the dispersion relation fails. """ print("\n*******************************") print("* Linear operator computation *") print("*******************************\n") # --- Step 1: symbolic variables --- omega = symbols("omega") if self.dim == 1: kvars = [symbols("kx")] space_vars = [self.x] elif self.dim == 2: kvars = symbols("kx ky") space_vars = [self.x, self.y] else: raise ValueError("Only 1D and 2D are supported.") kdict = dict(zip(space_vars, kvars)) self.k_symbols = kvars # Plane wave expression phase = sum(k * x for k, x in zip(kvars, space_vars)) - omega * self.t plane_wave = exp(I * phase) # --- Step 2: build lhs expression from linear terms --- lhs = 0 for deriv, coeff in self.linear_terms.items(): if isinstance(deriv, Derivative): total_factor = 1 for var, n in deriv.variable_count: if var == self.t: total_factor *= (-I * omega)**n elif var in kdict: total_factor *= (I * kdict[var])**n else: raise ValueError(f"Unknown variable {var} in derivative") lhs += coeff * total_factor * plane_wave elif deriv == self.u: lhs += coeff * plane_wave else: raise ValueError(f"Unsupported linear term: {deriv}") # --- Step 3: dispersion relation --- equation = simplify(lhs / plane_wave) print("\nCharacteristic equation before symbol treatment:") pprint(equation) print("\n--- Symbolic symbol analysis ---") symb_omega = 0 symb_k = 0 for coeff, symbol in self.symbol_terms: if symbol.has(omega): # Ajouter directement les termes dépendant de omega symb_omega += coeff * symbol elif any(symbol.has(k) for k in self.k_symbols): symb_k += coeff * symbol.subs(dict(zip(symbol.free_symbols, self.k_symbols))) print(f"symb_omega: {symb_omega}") print(f"symb_k: {symb_k}") equation = equation + symb_omega + symb_k print("\nRaw characteristic equation:") pprint(equation) # Temporal derivative order detection try: poly_eq = Eq(equation, 0) poly = poly_eq.lhs.as_poly(omega) self.temporal_order = poly.degree() if poly else 0 except Exception as e: warnings.warn(f"Could not determine temporal order: {e}", RuntimeWarning) self.temporal_order = 0 print(f"Temporal order from dispersion relation: {self.temporal_order}") print('self.pseudo_terms = ', self.pseudo_terms) if self.pseudo_terms: coeff_time = 1 for term, coeff in self.linear_terms.items(): if isinstance(term, Derivative) and any(var == self.t for var, _ in term.variable_count): coeff_time = coeff print(f"✅ Time derivative coefficient detected: {coeff_time}") self.psi_ops = [] for coeff, sym_expr in self.pseudo_terms: # expr est le Sympy expr. différentiel, var_x la liste [x] ou [x,y] psi = PseudoDifferentialOperator(sym_expr / coeff_time, self.spatial_vars, self.u, mode='symbol') self.psi_ops.append((coeff, psi)) else: dispersion = solve(Eq(equation, 0), omega) if not dispersion: raise ValueError("No solution found for omega") print("\n--- Solutions found ---") pprint(dispersion) if self.temporal_order == 2: omega_expr = simplify(sqrt(dispersion[0]**2)) self.omega_symbolic = omega_expr self.omega = lambdify(self.k_symbols, omega_expr, "numpy") self.L_symbolic = -omega_expr**2 else: self.L_symbolic = -I * dispersion[0] self.L = lambdify(self.k_symbols, self.L_symbolic, "numpy") print("\n--- Final linear operator ---") pprint(self.L_symbolic) def linear_rhs(self, u, is_v=False): """ Apply the linear operator (in Fourier space) to the field u or v. Parameters ---------- u : np.ndarray Input solution array. is_v : bool Whether to apply the operator to v instead of u. Returns ------- np.ndarray Result of applying the linear operator. """ if self.dim == 1: self.symbol_u = np.array(self.L(self.KX), dtype=np.complex128) self.symbol_v = self.symbol_u # même opérateur pour u et v elif self.dim == 2: self.symbol_u = np.array(self.L(self.KX, self.KY), dtype=np.complex128) self.symbol_v = self.symbol_u u_hat = self.fft(u) u_hat *= self.symbol_v if is_v else self.symbol_u u_hat *= self.dealiasing_mask return self.ifft(u_hat) def setup(self, Lx, Ly=None, Nx=None, Ny=None, Lt=1.0, Nt=100, boundary_condition='periodic', initial_condition=None, initial_velocity=None, n_frames=100): """ Configure the spatial/temporal grid and initialize the solution field. This method sets up the computational domain, initializes spatial and temporal grids, applies boundary conditions, and prepares symbolic and numerical operators. It also performs essential analyses such as: - CFL condition verification (for stability) - Symbol analysis (e.g., dispersion relation, regularity) - Wave propagation analysis for second-order equations If pseudo-differential operators (ψOp) are present, symbolic analysis is skipped in favor of interactive exploration via `interactive_symbol_analysis`. Parameters ---------- Lx : float Size of the spatial domain along x-axis. Ly : float, optional Size of the spatial domain along y-axis (for 2D problems). Nx : int Number of spatial points along x-axis. Ny : int, optional Number of spatial points along y-axis (for 2D problems). Lt : float, default=1.0 Total simulation time. Nt : int, default=100 Number of time steps. initial_condition : callable Function returning the initial state u(x, 0) or u(x, y, 0). initial_velocity : callable, optional Function returning the initial time derivative ∂ₜu(x, 0) or ∂ₜu(x, y, 0), required for second-order equations. n_frames : int, default=100 Number of time frames to store during simulation for visualization or output. Raises ------ ValueError If mandatory parameters are missing (e.g., Nx not given in 1D, Ly/Ny not given in 2D). Notes ----- - The spatial discretization assumes periodic boundary conditions by default. - Fourier transforms are computed using real-to-complex FFTs (`scipy.fft.fft`, `fft2`). - Frequency arrays (`KX`, `KY`) are defined following standard spectral conventions. - Dealiasing is applied using a sharp cutoff filter at a fraction of the maximum frequency. - For second-order equations, initial acceleration is derived from the governing operator. - Symbolic analysis includes plotting of the symbol's real/imaginary/absolute values, wavefront propagation, and dispersion relation. See Also -------- setup_1D : Sets up internal variables for one-dimensional problems. setup_2D : Sets up internal variables for two-dimensional problems. initialize_conditions : Applies initial data and enforces compatibility. check_cfl_condition : Verifies time step against stability constraints. plot_symbol : Visualizes the linear operator’s symbol in frequency space. analyze_wave_propagation : Analyzes group velocity and wavefront dynamics. interactive_symbol_analysis : Interactive tools for ψOp-based equations. """ # Temporal parameters self.Lt, self.Nt = Lt, Nt self.dt = Lt / Nt self.n_frames = n_frames self.frames = [] self.initial_condition = initial_condition self.boundary_condition = boundary_condition if self.boundary_condition == 'dirichlet' and not self.has_psi: raise ValueError( "Dirichlet boundary conditions require the equation to be defined via a pseudo-differential operator (psiOp). " "Please provide an equation involving psiOp for non-periodic boundary treatment." ) # Dimension checks if self.dim == 1: if Nx is None: raise ValueError("Nx must be specified in 1D.") self.setup_1D(Lx, Nx) else: if None in (Ly, Ny): raise ValueError("In 2D, Ly and Ny must be provided.") self.setup_2D(Lx, Ly, Nx, Ny) # Initialization of solution and velocities if not self.is_stationary: self.initialize_conditions(initial_condition, initial_velocity) # Symbol analysis if present if self.has_psi: print("⚠️ For psiOp, use interactive_symbol_analysis.") else: if self.L_symbolic == 0: print("⚠️ Linear operator is null.") else: self.check_cfl_condition() self.check_symbol_conditions() self.plot_symbol() if self.temporal_order == 2: self.analyze_wave_propagation() def setup_1D(self, Lx, Nx): """ Configure internal variables for one-dimensional (1D) problems. This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution. It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2]. Parameters ---------- Lx : float Physical size of the spatial domain along the x-axis. Nx : int Number of grid points in the x-direction. Attributes Set -------------- - self.Lx : float Size of the spatial domain. - self.Nx : int Number of spatial points. - self.x_grid : np.ndarray 1D array of spatial coordinates. - self.X : np.ndarray Alias to `self.x_grid`, used in physical space computations. - self.kx : np.ndarray Array of wavenumbers corresponding to the Fourier transform. - self.KX : np.ndarray Alias to `self.kx`, used in frequency space computations. - self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations. - self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(k) · dt). - self.omega_val : np.ndarray Frequency values ω(k) = Re[√(L(k))] used in second-order time stepping. - self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(k)·dt for dispersive propagation. - self.inv_omega : np.ndarray Inverse of ω(k), used to avoid division-by-zero in time stepping. Notes ----- - Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`. - For second-order equations, the dispersion relation ω(k) is extracted from the linear operator L(k). See Also -------- setup_2D : Equivalent setup for two-dimensional problems. prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation. setup_omega_terms : Sets up terms involving ω(k) for second-order evolution. """ self.Lx, self.Nx = Lx, Nx self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False) self.X = self.x_grid self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx) self.KX = self.kx # Dealiasing mask k_max = self.dealiasing_ratio * np.max(np.abs(self.kx)) self.dealiasing_mask = (np.abs(self.KX) <= k_max) # Preparation of symbol or linear operator if self.has_psi: self.prepare_symbol_tables() else: L_vals = np.array(self.L(self.KX), dtype=np.complex128) self.exp_L = np.exp(L_vals * self.dt) if self.temporal_order == 2: omega_val = self.omega(self.KX) self.setup_omega_terms(omega_val) def setup_2D(self, Lx, Ly, Nx, Ny): """ Configure internal variables for two-dimensional (2D) problems. This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution. It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2] × [-Ly/2, Ly/2]. Parameters ---------- Lx : float Physical size of the spatial domain along the x-axis. Ly : float Physical size of the spatial domain along the y-axis. Nx : int Number of grid points along the x-direction. Ny : int Number of grid points along the y-direction. Attributes Set -------------- - self.Lx, self.Ly : float Size of the spatial domain in each direction. - self.Nx, self.Ny : int Number of spatial points in each direction. - self.x_grid, self.y_grid : np.ndarray 1D arrays of spatial coordinates in x and y directions. - self.X, self.Y : np.ndarray 2D meshgrids of spatial coordinates for physical space computations. - self.kx, self.ky : np.ndarray Arrays of wavenumbers corresponding to Fourier transforms in x and y directions. - self.KX, self.KY : np.ndarray Meshgrids of wavenumbers used in frequency space computations. - self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations. - self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(kx, ky) · dt). - self.omega_val : np.ndarray Frequency values ω(kx, ky) = Re[√(L(kx, ky))] used in second-order time stepping. - self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(kx, ky)·dt for dispersive propagation. - self.inv_omega : np.ndarray Inverse of ω(kx, ky), used to avoid division-by-zero in time stepping. Notes ----- - Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`. - For second-order equations, the dispersion relation ω(kx, ky) is extracted from the linear operator L(kx, ky). See Also -------- setup_1D : Equivalent setup for one-dimensional problems. prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation. setup_omega_terms : Sets up terms involving ω(kx, ky) for second-order evolution. """ self.Lx, self.Ly = Lx, Ly self.Nx, self.Ny = Nx, Ny self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False) self.y_grid = np.linspace(-Ly/2, Ly/2, Ny, endpoint=False) self.X, self.Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij') self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx) self.ky = 2 * np.pi * fftfreq(Ny, d=Ly / Ny) self.KX, self.KY = np.meshgrid(self.kx, self.ky, indexing='ij') # Dealiasing mask kx_max = self.dealiasing_ratio * np.max(np.abs(self.kx)) ky_max = self.dealiasing_ratio * np.max(np.abs(self.ky)) self.dealiasing_mask = (np.abs(self.KX) <= kx_max) & (np.abs(self.KY) <= ky_max) # Preparation of symbol or linear operator if self.has_psi: self.prepare_symbol_tables() else: L_vals = self.L(self.KX, self.KY) self.exp_L = np.exp(L_vals * self.dt) if self.temporal_order == 2: omega_val = self.omega(self.KX, self.KY) self.setup_omega_terms(omega_val) def setup_omega_terms(self, omega_val): """ Initialize terms derived from the angular frequency ω for time evolution. This private method precomputes and stores key trigonometric and inverse quantities based on the dispersion relation ω(k), used in second-order time integration schemes. These values are essential for solving wave-like equations with dispersive behavior: cos(ω·dt), sin(ω·dt), 1/ω The inverse frequency is computed safely to avoid division by zero. Parameters ---------- omega_val : np.ndarray Array of angular frequency values ω(k) evaluated at discrete wavenumbers. Can be one-dimensional (1D) or two-dimensional (2D) depending on spatial dimension. Attributes Set -------------- - self.omega_val : np.ndarray Copy of the input angular frequency array. - self.cos_omega_dt : np.ndarray Cosine of ω(k) multiplied by time step: cos(ω(k) · dt). - self.sin_omega_dt : np.ndarray Sine of ω(k) multiplied by time step: sin(ω(k) · dt). - self.inv_omega : np.ndarray Inverse of ω(k), with zeros where ω(k) == 0 to avoid division by zero. Notes ----- - This method is typically called during setup when solving second-order PDEs involving dispersive waves (e.g., Klein-Gordon, Schrödinger, or water wave equations). - The safe computation of 1/ω ensures numerical stability even when low frequencies are present. - These precomputed arrays are used in spectral propagators for accurate time stepping. See Also -------- setup_1D : Sets up internal variables for one-dimensional problems. setup_2D : Sets up internal variables for two-dimensional problems. solve : Time integration using the computed frequency terms. """ self.omega_val = omega_val self.cos_omega_dt = np.cos(omega_val * self.dt) self.sin_omega_dt = np.sin(omega_val * self.dt) self.inv_omega = np.zeros_like(omega_val) nonzero = omega_val != 0 self.inv_omega[nonzero] = 1.0 / omega_val[nonzero] def evaluate_source_at_t0(self): """ Evaluate source terms at initial time t = 0 over the spatial grid. This private method computes the total contribution of all source terms at the initial time, evaluated across the entire spatial domain. It supports both one-dimensional (1D) and two-dimensional (2D) configurations. Returns ------- np.ndarray A numpy array representing the evaluated source term at t=0: - In 1D: Shape (Nx,), evaluated at each x in `self.x_grid`. - In 2D: Shape (Nx, Ny), evaluated at each (x, y) pair in the grid. Notes ----- - The symbolic expressions in `self.source_terms` are substituted with numerical values at t=0. - In 1D, each term is evaluated at (t=0, x=x_val). - In 2D, each term is evaluated at (t=0, x=x_val, y=y_val). - Evaluated using SymPy's `evalf()` to ensure numeric conversion. - This method assumes that the source terms have already been lambdified or are compatible with symbolic substitution. See Also -------- setup : Initializes the spatial grid and source terms. solve : Uses this evaluation during the first time step. """ if self.dim == 1: # Evaluation on the 1D spatial grid return np.array([ sum(term.subs(self.t, 0).subs(self.x, x_val).evalf() for term in self.source_terms) for x_val in self.x_grid ], dtype=np.float64) else: # Evaluation on the 2D spatial grid return np.array([ [sum(term.subs({self.t: 0, self.x: x_val, self.y: y_val}).evalf() for term in self.source_terms) for y_val in self.y_grid] for x_val in self.x_grid ], dtype=np.float64) def initialize_conditions(self, initial_condition, initial_velocity): """ Initialize the solution and velocity fields at t = 0. This private method sets up the initial state of the solution `u_prev` and, if applicable, the time derivative (velocity) `v_prev` for second-order evolution equations. For second-order equations, it also computes the backward-in-time value `u_prev2` needed by the Leap-Frog method. The acceleration at t = 0 is computed from: ∂ₜ²u = L(u) + N(u) + f(x, t=0) where L is the linear operator, N is the nonlinear term, and f is the source term. Parameters ---------- initial_condition : callable Function returning the initial condition u(x, 0) or u(x, y, 0). initial_velocity : callable or None Function returning the initial velocity ∂ₜu(x, 0) or ∂ₜu(x, y, 0). Required for second-order equations; ignored otherwise. Raises ------ ValueError If `initial_velocity` is not provided for second-order equations. Notes ----- - Applies periodic boundary conditions after setting initial data. - Stores a copy of the initial state in `self.frames` for visualization/output. - In second-order systems, initializes `self.u_prev2` using a Taylor expansion: u_prev2 = u_prev - dt * v_prev + 0.5 * dt² * (∂ₜ²u) See Also -------- apply_boundary : Enforces periodic boundary conditions on the solution field. psiOp_apply : Computes pseudo-differential operator action for acceleration. linear_rhs : Evaluates linear part of the equation in Fourier space. apply_nonlinear : Handles nonlinear terms with spectral differentiation. evaluate_source_at_t0 : Evaluates source terms at the initial time. """ # Initial condition if self.dim == 1: self.u_prev = initial_condition(self.X) else: self.u_prev = initial_condition(self.X, self.Y) self.apply_boundary(self.u_prev) # Initial velocity (second order) if self.temporal_order == 2: if initial_velocity is None: raise ValueError("Initial velocity is required for second-order equations.") if self.dim == 1: self.v_prev = initial_velocity(self.X) else: self.v_prev = initial_velocity(self.X, self.Y) self.u0 = np.copy(self.u_prev) self.v0 = np.copy(self.v_prev) # Calculation of u_prev2 (initial acceleration) if not hasattr(self, 'u_prev2'): if self.has_psi: acc0 = self.apply_psiOp(self.u_prev) else: acc0 = self.linear_rhs(self.u_prev, is_v=False) rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False) acc0 += rhs_nl if hasattr(self, 'source_terms') and self.source_terms: acc0 += self.evaluate_source_at_t0() self.u_prev2 = self.u_prev - self.dt * self.v_prev + 0.5 * self.dt**2 * acc0 self.frames = [self.u_prev.copy()] def apply_boundary(self, u): """ Apply boundary conditions to the solution array based on the specified type. This method supports two types of boundary conditions: - 'periodic': Enforces periodicity by copying opposite boundary values. - 'dirichlet': Sets all boundary values to zero (homogeneous Dirichlet condition). Parameters ---------- u : np.ndarray The solution array representing the field values on a spatial grid. In 1D, shape must be (Nx,). In 2D, shape must be (Nx, Ny). Raises ------ ValueError If `self.boundary_condition` is not one of {'periodic', 'dirichlet'}. Notes ----- - For 'periodic': * In 1D: u[0] = u[-2], u[-1] = u[1] * In 2D: First and last rows/columns are set equal to their neighbors. - For 'dirichlet': * All boundary points are explicitly set to zero. """ if self.boundary_condition == 'periodic': if self.dim == 1: u[0] = u[-2] u[-1] = u[1] elif self.dim == 2: u[0, :] = u[-2, :] u[-1, :] = u[1, :] u[:, 0] = u[:, -2] u[:, -1] = u[:, 1] elif self.boundary_condition == 'dirichlet': if self.dim == 1: u[0] = 0 u[-1] = 0 elif self.dim == 2: u[0, :] = 0 u[-1, :] = 0 u[:, 0] = 0 u[:, -1] = 0 else: raise ValueError( f"Invalid boundary condition '{self.boundary_condition}'. " "Supported types are 'periodic' and 'dirichlet'." ) def apply_nonlinear(self, u, is_v=False): """ Apply nonlinear terms to the solution using spectral differentiation with dealiasing. This method evaluates all nonlinear terms present in the PDE by substituting spatial derivatives with their spectral approximations computed via FFT. The dealiasing mask ensures numerical stability by removing high-frequency components that could lead to aliasing errors. Parameters ---------- u : numpy.ndarray Current solution array on the spatial grid. is_v : bool If True, evaluates nonlinear terms for the velocity field v instead of u. Returns: numpy.ndarray: Array representing the contribution of nonlinear terms multiplied by dt. Notes: - In 1D, computes ∂ₓu via FFT and substitutes any derivative term in the nonlinear expressions. - In 2D, computes ∂ₓu and ∂ᵧu via FFT and performs similar substitutions. - Uses lambdify to evaluate symbolic nonlinear expressions numerically. - Derivatives are replaced symbolically with 'u_x' and 'u_y' before evaluation. """ if not self.nonlinear_terms: return np.zeros_like(u, dtype=np.complex128) nonlinear_term = np.zeros_like(u, dtype=np.complex128) if self.dim == 1: u_hat = self.fft(u) u_hat *= self.dealiasing_mask u = self.ifft(u_hat) u_x_hat = (1j * self.KX) * u_hat u_x = self.ifft(u_x_hat) for term in self.nonlinear_terms: term_replaced = term if term.has(Derivative): for deriv in term.atoms(Derivative): if deriv.args[1][0] == self.x: term_replaced = term_replaced.subs(deriv, symbols('u_x')) term_func = lambdify((self.t, self.x, self.u_eq, 'u_x'), term_replaced, 'numpy') if is_v: nonlinear_term += term_func(0, self.X, self.v_prev, u_x) else: nonlinear_term += term_func(0, self.X, u, u_x) elif self.dim == 2: u_hat = self.fft(u) u_hat *= self.dealiasing_mask u = self.ifft(u_hat) u_x_hat = (1j * self.KX) * u_hat u_y_hat = (1j * self.KY) * u_hat u_x = self.ifft(u_x_hat) u_y = self.ifft(u_y_hat) for term in self.nonlinear_terms: term_replaced = term if term.has(Derivative): for deriv in term.atoms(Derivative): if deriv.args[1][0] == self.x: term_replaced = term_replaced.subs(deriv, symbols('u_x')) elif deriv.args[1][0] == self.y: term_replaced = term_replaced.subs(deriv, symbols('u_y')) term_func = lambdify((self.t, self.x, self.y, self.u_eq, 'u_x', 'u_y'), term_replaced, 'numpy') if is_v: nonlinear_term += term_func(0, self.X, self.Y, self.v_prev, u_x, u_y) else: nonlinear_term += term_func(0, self.X, self.Y, u, u_x, u_y) else: raise ValueError("Unsupported spatial dimension.") return nonlinear_term * self.dt def prepare_symbol_tables(self): """ Precompute and store evaluated pseudo-differential operator symbols for spectral methods. This method evaluates all pseudo-differential operators (ψOp) present in the PDE over the spatial and frequency grids, scales them by their respective coefficients, and combines them into a single composite symbol used in time-stepping and inversion. The evaluation is performed via the `evaluate` method of each PseudoDifferentialOperator, which computes p(x, ξ) or p(x, y, ξ, η) numerically over the current grid configuration. Side Effects: self.precomputed_symbols : list of (coeff, symbol_array) Each tuple contains a coefficient and its evaluated symbol on the grid. self.combined_symbol : np.ndarray Sum of all scaled symbol arrays: ∑(coeffₖ * ψₖ(x, ξ)) Raises: ValueError: If the spatial dimension is not 1D or 2D. """ self.precomputed_symbols = [] self.combined_symbol = 0 for coeff, psi in self.psi_ops: if self.dim == 1: raw = psi.evaluate(self.X, None, self.KX, None) elif self.dim == 2: raw = psi.evaluate(self.X, self.Y, self.KX, self.KY) else: raise ValueError('Unsupported spatial dimension.') raw_flat = raw.flatten() converted = np.array([complex(N(val)) for val in raw_flat], dtype=np.complex128) raw_eval = converted.reshape(raw.shape) self.precomputed_symbols.append((coeff, raw_eval)) self.combined_symbol = sum((coeff * sym for coeff, sym in self.precomputed_symbols)) self.combined_symbol = np.array(self.combined_symbol, dtype=np.complex128) def total_symbol_expr(self): """ Compute the total pseudo-differential symbol expression from all pseudo_terms. This method constructs the full symbol of the pseudo-differential operator by summing up all coefficient-weighted symbolic expressions. The result is cached in self.symbol_expr to avoid recomputation. Returns: sympy.Expr: The combined symbol expression, representing the full pseudo-differential operator in symbolic form. Example: Given pseudo_terms = [(2, ξ²), (1, x·ξ)], this returns 2·ξ² + x·ξ. """ if not hasattr(self, '_symbol_expr'): self.symbol_expr = sum(coeff * expr for coeff, expr in self.pseudo_terms) return self.symbol_expr def build_symbol_func(self, expr): """ Build a numerical evaluation function from a symbolic pseudo-differential operator expression. This method converts a symbolic expression representing a pseudo-differential operator into a callable NumPy-compatible function. The function accepts spatial and frequency variables depending on the dimensionality of the problem. Parameters ---------- expr : sympy expression A SymPy expression representing the symbol of the pseudo-differential operator. It may depend on spatial variables (x, y) and frequency variables (xi, eta). Returns: function : A lambdified function that takes: - In 1D: `(x, xi)` — spatial coordinate and frequency. - In 2D: `(x, y, xi, eta)` — spatial coordinates and frequencies. Returns a NumPy array of evaluated symbol values over input grids. Notes: - Uses `lambdify` from SymPy with the `'numpy'` backend for efficient vectorized evaluation. - Real variable assumptions are enforced to ensure proper behavior in numerical contexts. - Used internally by methods like `apply_psiOp`, `evaluate`, and visualization tools. """ if self.dim == 1: x, xi = symbols('x xi', real=True) return lambdify((x, xi), expr, 'numpy') else: x, y, xi, eta = symbols('x y xi eta', real=True) return lambdify((x, y, xi, eta), expr, 'numpy') def apply_psiOp(self, u): """ Apply the pseudo-differential operator to the input field u. This method dispatches the application of the pseudo-differential operator based on: - Whether the symbol is spatially dependent (x/y) - The boundary condition in use (periodic or dirichlet) Supported operations: - Constant-coefficient symbols: applied via Fourier multiplication. - Spatially varying symbols: applied via Kohn–Nirenberg quantization. - Dirichlet boundary conditions: handled with non-periodic convolution-like quantization. Dispatch Logic:\n if not self.is_spatial: u ↦ Op(p)(D) ⋅ u = 𝓕⁻¹[ p(ξ) ⋅ 𝓕(u) ]\n elif periodic: u ↦ Op(p)(x,D) ⋅ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ based of FFT (quicker)\n elif dirichlet: u ↦ Op(p)(x,D) ⋅ u ≈ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ (slower)\n Parameters ---------- u : np.ndarray Input field to which the operator is applied. Should be 1D or 2D depending on the problem dimension. Returns: np.ndarray: Result of applying the pseudo-differential operator to u. Raises: ValueError: If an unsupported boundary condition is specified. """ if not self.is_spatial: return self.apply_psiOp_constant(u) elif self.boundary_condition == 'periodic': return self.apply_psiOp_kohn_nirenberg_fft(u) elif self.boundary_condition == 'dirichlet': return self.apply_psiOp_kohn_nirenberg_nonperiodic(u) else: raise ValueError(f"Invalid boundary condition '{self.boundary_condition}'") def apply_psiOp_constant(self, u): """ Apply a constant-coefficient pseudo-differential operator in Fourier space. This method assumes the symbol is diagonal in the Fourier basis and acts as a multiplication operator. It performs the operation: (ψu)(x) = 𝓕⁻¹[ -σ(k) · 𝓕[u](k) ] where: - σ(k) is the combined pseudo-differential operator symbol - 𝓕 denotes the forward Fourier transform - 𝓕⁻¹ denotes the inverse Fourier transform The dealiasing mask is applied before returning to physical space. Parameters ---------- u : np.ndarray Input function in physical space (real-valued or complex-valued) Returns: np.ndarray : Result of applying the pseudo-differential operator to u, same shape as input """ u_hat = self.fft(u) u_hat *= -self.combined_symbol u_hat *= self.dealiasing_mask return self.ifft(u_hat) def apply_psiOp_kohn_nirenberg_fft(self, u): """ Apply a pseudo-differential operator using the Kohn–Nirenberg quantization in Fourier space. This method evaluates the action of a pseudo-differential operator defined by the total symbol, computed from all psiOp terms in the equation. It uses the fast Fourier transform (FFT) for efficiency in periodic domains. Parameters ---------- u : np.ndarray Input function in real space to which the operator is applied. Returns: np.ndarray: Resulting function after applying the pseudo-differential operator. Process: 1. Compute the total symbolic expression of the pseudo-differential operator. 2. Build a callable numerical function from the symbol. 3. Evaluate Op(p)(u) via the Kohn–Nirenberg quantization using FFT. Note: - Assumes periodic boundary conditions. - The returned result is the negative of the standard definition due to PDE sign conventions. """ total_symbol = self.total_symbol_expr() symbol_func = self.build_symbol_func(total_symbol) return -self.kohn_nirenberg_fft(u_vals=u, symbol_func=symbol_func) def apply_psiOp_kohn_nirenberg_nonperiodic(self, u): """ Apply a pseudo-differential operator using the Kohn–Nirenberg quantization on non-periodic domains. This method evaluates the action of a pseudo-differential operator Op(p) on a function u via the Kohn–Nirenberg representation. It supports both 1D and 2D cases and uses spatial and frequency grids to evaluate the operator symbol p(x, ξ). The operator symbol p(x, ξ) is extracted from the PDE and evaluated numerically using `_total_symbol_expr` and `_build_symbol_func`. Parameters ---------- u : np.ndarray Input function (real space) to which the operator is applied. Returns: np.ndarray: Result of applying Op(p) to u in real space. Notes: - For 1D: p(x, ξ) is evaluated over x_grid and xi_grid. - For 2D: p(x, y, ξ, η) is evaluated over (x_grid, y_grid) and (xi_grid, eta_grid). - The result is computed using `kohn_nirenberg_nonperiodic`, which handles non-periodic boundary conditions. """ total_symbol = self.total_symbol_expr() symbol_func = self.build_symbol_func(total_symbol) if self.dim == 1: return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=self.x_grid, xi_grid=self.kx, symbol_func=symbol_func) else: return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=symbol_func) def step_order1_with_psi(self, source_contribution): """ Perform one time step of a first-order evolution using a pseudo-differential operator. This method updates the solution field using an exponential integrator or explicit Euler scheme, depending on boundary conditions and the structure of the pseudo-differential symbol. It supports: - Linear dynamics via pseudo-differential operator L (possibly nonlocal) - Nonlinear terms computed via spectral differentiation - External source contributions The update follows **three distinct computational paths**: 1. **Periodic boundaries + diagonalizable symbol** Symbol is constant in space → use direct Fourier-based exponential integrator: uₙ₊₁ = e⁻ᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(−LΔt) ⋅ (N(uₙ) + F) 2. **Non-diagonalizable but spatially uniform symbol** General exponential time differencing of order 1: uₙ₊₁ = eᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(LΔt) ⋅ (N(uₙ) + F) 3. **Spatially varying symbol** No frequency diagonalization available → use explicit Euler: uₙ₊₁ = uₙ + Δt ⋅ (L(uₙ) + N(uₙ) + F) where: L(uₙ) = linear part via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term Δt = time step size φ₁(z) = (eᶻ − 1)/z (with safe handling near z=0) Boundary conditions are applied after each update to ensure consistency. Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev. Returns: np.ndarray: Updated solution array after one time step. """ # Handling null source if np.isscalar(source_contribution): source = np.zeros_like(self.u_prev) else: source = source_contribution def spectral_filter(u, cutoff=0.8): if u.ndim == 1: u_hat = self.fft(u) N = len(u) k = fftfreq(N) mask = np.exp(-(k / cutoff)**8) return self.ifft(u_hat * mask).real elif u.ndim == 2: u_hat = self.fft(u) Ny, Nx = u.shape ky = fftfreq(Ny)[:, None] kx = fftfreq(Nx)[None, :] k_squared = kx**2 + ky**2 mask = np.exp(-(np.sqrt(k_squared) / cutoff)**8) return self.ifft(u_hat * mask).real else: raise ValueError("Only 1D and 2D arrays are supported.") # Recalculate symbol if necessary if self.is_spatial: self.prepare_symbol_tables() # Recalculates self.combined_symbol # Case with FFT (symbol diagonalizable in Fourier space) if self.boundary_condition == 'periodic' and not self.is_spatial: u_hat = self.fft(self.u_prev) u_hat *= np.exp(-self.dt * self.combined_symbol) u_hat *= self.dealiasing_mask u_symb = self.ifft(u_hat) u_nl = self.apply_nonlinear(self.u_prev) u_new = u_symb + u_nl + source else: if not self.is_spatial: # General case with ETD1 u_nl = self.apply_nonlinear(self.u_prev) # Calculation of exp(dt * L) and phi1(dt * L) L_vals = self.combined_symbol # Uses the updated symbol exp_L = np.exp(-self.dt * L_vals) phi1_L = (exp_L - 1.0) / (self.dt * L_vals) phi1_L[np.isnan(phi1_L)] = 1.0 # Handling division by zero # Fourier transform u_hat = self.fft(self.u_prev) u_nl_hat = self.fft(u_nl) source_hat = self.fft(source) # Assembling the solution in Fourier space u_hat_new = exp_L * u_hat + self.dt * phi1_L * (u_nl_hat + source_hat) u_new = self.ifft(u_hat_new) else: # if the symbol depends on spatial variables : Euler method Lu_prev = self.apply_psiOp(self.u_prev) u_nl = self.apply_nonlinear(self.u_prev) u_new = self.u_prev + self.dt * (Lu_prev + u_nl + source) u_new = spectral_filter(u_new, cutoff=self.dealiasing_ratio) # Applying boundary conditions self.apply_boundary(u_new) return u_new def step_order2_with_psi(self, source_contribution): """ Perform one time step of a second-order time evolution using a pseudo-differential operator. This method updates the solution field using a second-order accurate scheme suitable for wave-like equations. The update includes contributions from: - Linear dynamics via a pseudo-differential operator (e.g., dispersion or stiffness) - Nonlinear terms computed via spectral differentiation - External source contributions Discretization follows a leapfrog-style finite difference in time: uₙ₊₁ = 2uₙ − uₙ₋₁ + Δt² ⋅ (L(uₙ) + N(uₙ) + F) where: L(uₙ) = linear part evaluated via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term at current time step Δt = time step size Boundary conditions are applied after each update to ensure consistency. Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev. Returns: np.ndarray: Updated solution array after one time step. """ Lu_prev = self.apply_psiOp(self.u_prev) rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False) u_new = 2 * self.u_prev - self.u_prev2 + self.dt ** 2 * (Lu_prev + rhs_nl + source_contribution) self.apply_boundary(u_new) self.u_prev2 = self.u_prev self.u_prev = u_new self.u = u_new return u_new def solve(self): """ Solve the partial differential equation numerically using spectral methods. This method evolves the solution in time using a combination of: - Fourier-based linear evolution (with dealiasing) - Nonlinear term handling via pseudo-spectral evaluation - Support for pseudo-differential operators (psiOp) - Source terms and boundary conditions The solver supports: - 1D and 2D spatial domains - First and second-order time evolution - Periodic and Dirichlet boundary conditions - Time-stepping schemes: default, ETD-RK4 Returns: list[np.ndarray]: A list of solution arrays at each saved time frame. Side Effects: - Updates self.frames: stores solution snapshots - Updates self.energy_history: records total energy if enabled Algorithm Overview: For each time step: 1. Evaluate source contributions (if any) 2. Apply time evolution: - Order 1: - With psiOp: uses step_order1_with_psi - With ETD-RK4: exponential time differencing - Default: linear + nonlinear update - Order 2: - With psiOp: uses step_order2_with_psi - With ETD-RK4: second-order exponential scheme - Default: second-order leapfrog-style update 3. Enforce boundary conditions 4. Save solution snapshot periodically 5. Record energy (for second-order systems without psiOp) """ print('\n*******************') print('* Solving the PDE *') print('*******************\n') save_interval = max(1, self.Nt // self.n_frames) self.energy_history = [] for step in range(self.Nt): if hasattr(self, 'source_terms') and self.source_terms: source_contribution = np.zeros_like(self.X, dtype=np.float64) for term in self.source_terms: try: if self.dim == 1: source_func = lambdify((self.t, self.x), term, 'numpy') source_contribution += source_func(step * self.dt, self.X) elif self.dim == 2: source_func = lambdify((self.t, self.x, self.y), term, 'numpy') source_contribution += source_func(step * self.dt, self.X, self.Y) except Exception as e: print(f'Error evaluating source term {term}: {e}') else: source_contribution = 0 if self.temporal_order == 1: if self.has_psi: u_new = self.step_order1_with_psi(source_contribution) elif hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4': u_new = self.step_ETD_RK4(self.u_prev) else: u_hat = self.fft(self.u_prev) u_hat *= self.exp_L u_hat *= self.dealiasing_mask u_lin = self.ifft(u_hat) u_nl = self.apply_nonlinear(u_lin) u_new = u_lin + u_nl + source_contribution self.apply_boundary(u_new) self.u_prev = u_new elif self.temporal_order == 2: if self.has_psi: u_new = self.step_order2_with_psi(source_contribution) else: if hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4': u_new, v_new = self.step_ETD_RK4_order2(self.u_prev, self.v_prev) else: u_hat = self.fft(self.u_prev) v_hat = self.fft(self.v_prev) u_new_hat = self.cos_omega_dt * u_hat + self.sin_omega_dt * self.inv_omega * v_hat v_new_hat = -self.omega_val * self.sin_omega_dt * u_hat + self.cos_omega_dt * v_hat u_new = self.ifft(u_new_hat) v_new = self.ifft(v_new_hat) u_nl = self.apply_nonlinear(self.u_prev, is_v=False) v_nl = self.apply_nonlinear(self.v_prev, is_v=True) u_new += (u_nl + source_contribution) * self.dt ** 2 / 2 v_new += (u_nl + source_contribution) * self.dt self.apply_boundary(u_new) self.apply_boundary(v_new) self.u_prev = u_new self.v_prev = v_new if step % save_interval == 0: self.frames.append(self.u_prev.copy()) if self.temporal_order == 2 and (not self.has_psi): E = self.compute_energy() self.energy_history.append(E) return self.frames def solve_stationary_psiOp(self, order=3): """ Solve stationary pseudo-differential equations of the form P[u] = f(x) or P[u] = f(x,y) using asymptotic inversion. This method computes the solution to a stationary (time-independent) pseudo-differential equation where the operator P is defined via symbolic expressions (psiOp). It constructs an asymptotic right inverse R such that P∘R ≈ Id, then applies it to the source term f using either direct Fourier multiplication (when the symbol is spatially independent) or Kohn–Nirenberg quantization (when spatial dependence is present). The inversion is based on the principal symbol of the operator and its asymptotic expansion up to the given order. Ellipticity of the symbol is checked numerically before inversion to ensure well-posedness. Parameters ---------- order : int, default=3 Order of the asymptotic expansion used to construct the right inverse of the pseudo-differential operator. method : str, optional Inversion strategy: - 'diagonal' (default): Fast approximate inversion using diagonal operators in frequency space. - 'full' : Pointwise exact inversion (slower but more accurate). Returns ------- ndarray The computed solution u(x) in 1D or u(x, y) in 2D as a NumPy array over the spatial grid. Raises ------ ValueError If no pseudo-differential operator (psiOp) is defined. If linear or nonlinear terms other than psiOp are present. If the symbol is not elliptic on the grid. If no source term is provided for the right-hand side. Notes ----- - The method assumes the problem is fully stationary: time derivatives must be absent. - Requires the equation to be purely pseudo-differential (no Op, Derivative, or nonlinear terms). - Symbol evaluation and inversion are dimension-aware (supports both 1D and 2D problems). - Supports optimization paths when the symbol does not depend on spatial variables. See Also -------- right_inverse_asymptotic : Constructs the asymptotic inverse of the pseudo-differential operator. kohn_nirenberg : Numerical implementation of general pseudo-differential operators. is_elliptic_numerically : Verifies numerical ellipticity of the symbol. """ print("\n*******************************") print("* Solving the stationnary PDE *") print("*******************************\n") print("boundary condition: ",self.boundary_condition) if not self.has_psi: raise ValueError("Only supports problems with psiOp.") if self.linear_terms or self.nonlinear_terms: raise ValueError("Stationary psiOp problems must be linear and purely pseudo-differential.") if self.boundary_condition not in ('periodic', 'dirichlet'): raise ValueError( "For stationary PDEs, boundary conditions must be explicitly defined. " "Supported types are 'periodic' and 'dirichlet'." ) if self.dim == 1: x = self.x xi = symbols('xi', real=True) spatial_vars = (x,) freq_vars = (xi,) X, KX = self.X, self.KX elif self.dim == 2: x, y = self.x, self.y xi, eta = symbols('xi eta', real=True) spatial_vars = (x, y) freq_vars = (xi, eta) X, Y, KX, KY = self.X, self.Y, self.KX, self.KY else: raise ValueError("Unsupported spatial dimension.") total_symbol = sum(coeff * psi.expr for coeff, psi in self.psi_ops) psi_total = PseudoDifferentialOperator(total_symbol, spatial_vars, mode='symbol') # Check ellipticity if self.dim == 1: is_elliptic = psi_total.is_elliptic_numerically(X, KX) else: is_elliptic = psi_total.is_elliptic_numerically((X[:, 0], Y[0, :]), (KX[:, 0], KY[0, :])) if not is_elliptic: raise ValueError("❌ The pseudo-differential symbol is not numerically elliptic on the grid.") print("✅ Elliptic pseudo-differential symbol: inversion allowed.") R_symbol = psi_total.right_inverse_asymptotic(order=order) print("Right inverse asymptotic symbol:") pprint(R_symbol, num_columns=150) if self.dim == 1: if R_symbol.has(x): R_func = lambdify((x, xi), R_symbol, modules='numpy') else: R_func = lambdify((xi,), R_symbol, modules='numpy') else: if R_symbol.has(x) or R_symbol.has(y): R_func = lambdify((x, y, xi, eta), R_symbol, modules='numpy') else: R_func = lambdify((xi, eta), R_symbol, modules='numpy') # Build rhs if self.source_terms: f_expr = sum(self.source_terms) used_vars = [v for v in spatial_vars if f_expr.has(v)] f_func = lambdify(used_vars, -f_expr, modules='numpy') if self.dim == 1: rhs = f_func(self.x_grid) if used_vars else np.zeros_like(self.x_grid) else: rhs = f_func(self.X, self.Y) if used_vars else np.zeros_like(self.X) elif self.initial_condition: raise ValueError("Initial condition should be None for stationnary equation.") else: raise ValueError("No source term provided to construct the right-hand side.") f_hat = self.fft(rhs) if self.boundary_condition == 'periodic': if self.dim == 1: if not R_symbol.has(x): print("⚡ Optimization: symbol independent of x — direct product in Fourier.") R_vals = R_func(self.KX) u_hat = R_vals * f_hat u = self.ifft(u_hat) else: print("⚙️ 1D Kohn-Nirenberg Quantification") x, xi = symbols('x xi', real=True) R_func = lambdify((x, xi), R_symbol, 'numpy') # Still 2 args for uniformity u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func) elif self.dim == 2: if not R_symbol.has(x) and not R_symbol.has(y): print("⚡ Optimization: Symbol independent of x and y — direct product in 2D Fourier.") R_vals = np.vectorize(R_func)(self.KX, self.KY) u_hat = R_vals * f_hat u = self.ifft(u_hat) else: print("⚙️ 2D Kohn-Nirenberg Quantification") x, xi, y, eta = symbols('x xi y eta', real=True) R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy') # Still 2 args for uniformity u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func) self.u = u return u elif self.boundary_condition == 'dirichlet': if self.dim == 1: x, xi = symbols('x xi', real=True) R_func = lambdify((x, xi), R_symbol, 'numpy') u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=X, xi_grid=KX, symbol_func=R_func) elif self.dim == 2: x, xi, y, eta = symbols('x xi y eta', real=True) R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy') u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=R_func) self.u = u return u else: raise ValueError( f"Invalid boundary condition '{self.boundary_condition}'. " "Supported types are 'periodic' and 'dirichlet'." ) def kohn_nirenberg_fft(self, u_vals, symbol_func, freq_window='gaussian', clamp=1e6, space_window=False): """ Numerically stable Kohn–Nirenberg quantization of a pseudo-differential operator. Applies the pseudo-differential operator Op(p) to the function f via the Kohn–Nirenberg quantization: [Op(p)f](x) = (1/(2π)^d) ∫ p(x, ξ) e^{ix·ξ} ℱ[f](ξ) dξ where p(x, ξ) is a symbol that may depend on both spatial variables x and frequency variables ξ. This method supports both 1D and 2D cases and includes optional smoothing techniques to improve numerical stability. Parameters ---------- u_vals : np.ndarray Spatial samples of the input function f(x) or f(x, y), defined on a uniform grid. symbol_func : callable A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array. freq_window : {'gaussian', 'hann', None}, optional Type of frequency-domain window to apply: - 'gaussian': smooth decay near high frequencies - 'hann': cosine-based tapering with hard cutoff - None: no frequency window applied clamp : float, optional Upper bound on the absolute value of the symbol. Prevents numerical blow-up from large values. space_window : bool, optional Whether to apply a spatial Gaussian window to suppress edge effects in physical space. Returns ------- np.ndarray The result of applying the pseudo-differential operator to f, returned as a real or complex array of the same shape as u_vals. Notes ----- - The implementation uses FFT-based quadrature of the inverse Fourier transform. - Symbol evaluation is vectorized over spatial and frequency grids. - Frequency and spatial windows help mitigate oscillatory behavior and aliasing. - In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η). """ # === Common setup === xg = self.x_grid dx = xg[1] - xg[0] if self.dim == 1: # === 1D case === # Frequency grid (shifted to center zero) Nx = self.Nx k = 2 * np.pi * fftshift(fftfreq(Nx, d=dx)) dk = k[1] - k[0] # Centered FFT of input f_shift = fftshift(u_vals) f_hat = self.fft(f_shift) * dx f_hat = fftshift(f_hat) # Build meshgrid for (x, ξ) X, K = np.meshgrid(xg, k, indexing='ij') # Evaluate the symbol p(x, ξ) P = symbol_func(X, K) # Optional: clamp extreme values P = np.clip(P, -clamp, clamp) # === Frequency-domain window === if freq_window == 'gaussian': sigma = 0.8 * np.max(np.abs(k)) W = np.exp(-(K / sigma) ** 4) P *= W elif freq_window == 'hann': W = 0.5 * (1 + np.cos(np.pi * K / np.max(np.abs(K)))) P *= W * (np.abs(K) < np.max(np.abs(K))) # === Optional spatial window === if space_window: x0 = (xg[0] + xg[-1]) / 2 L = (xg[-1] - xg[0]) / 2 S = np.exp(-((X - x0) / L) ** 2) P *= S # === Oscillatory kernel and integration === kernel = np.exp(1j * X * K) integrand = P * f_hat[None, :] * kernel # Approximate inverse Fourier integral u = np.sum(integrand, axis=1) * dk / (2 * np.pi) return u else: # === 2D case === yg = self.y_grid dy = yg[1] - yg[0] Nx, Ny = self.Nx, self.Ny # Frequency grids kx = 2 * np.pi * fftshift(fftfreq(Nx, d=dx)) ky = 2 * np.pi * fftshift(fftfreq(Ny, d=dy)) dkx = kx[1] - kx[0] dky = ky[1] - ky[0] # 2D FFT of f(x, y) f_shift = fftshift(u_vals) f_hat = self.fft(f_shift) * dx * dy f_hat = fftshift(f_hat) # Create 4D grids for broadcasting X, Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij') KX, KY = np.meshgrid(kx, ky, indexing='ij') Xb = X[:, :, None, None] Yb = Y[:, :, None, None] KXb = KX[None, None, :, :] KYb = KY[None, None, :, :] # Evaluate p(x, y, ξ, η) P_vals = symbol_func(Xb, Yb, KXb, KYb) P_vals = np.clip(P_vals, -clamp, clamp) # === Frequency windowing === if freq_window == 'gaussian': sigma_kx = 0.8 * np.max(np.abs(kx)) sigma_ky = 0.8 * np.max(np.abs(ky)) W_kx = np.exp(-(KXb / sigma_kx) ** 4) W_ky = np.exp(-(KYb / sigma_ky) ** 4) P_vals *= W_kx * W_ky elif freq_window == 'hann': Wx = 0.5 * (1 + np.cos(np.pi * KXb / np.max(np.abs(kx)))) Wy = 0.5 * (1 + np.cos(np.pi * KYb / np.max(np.abs(ky)))) mask_x = np.abs(KXb) < np.max(np.abs(kx)) mask_y = np.abs(KYb) < np.max(np.abs(ky)) P_vals *= Wx * Wy * mask_x * mask_y # === Optional spatial tapering === if space_window: x0 = (self.x_grid[0] + self.x_grid[-1]) / 2 y0 = (self.y_grid[0] + self.y_grid[-1]) / 2 Lx = (self.x_grid[-1] - self.x_grid[0]) / 2 Ly = (self.y_grid[-1] - self.y_grid[0]) / 2 S = np.exp(-((Xb - x0) / Lx) ** 2 - ((Yb - y0) / Ly) ** 2) P_vals *= S # === Oscillatory kernel and integration === phase = np.exp(1j * (Xb * KXb + Yb * KYb)) integrand = P_vals * phase * f_hat[None, None, :, :] # 2D Fourier inversion (numerical integration) u = np.sum(integrand, axis=(2, 3)) * dkx * dky / (2 * np.pi) ** 2 return u def kohn_nirenberg_nonperiodic(self, u_vals, x_grid, xi_grid, symbol_func, freq_window='gaussian', clamp=1e6, space_window=False): """ Numerically applies the Kohn–Nirenberg quantization of a pseudo-differential operator in a non-periodic setting. This method computes: [Op(p)u](x) = (1/(2π)^d) ∫ p(x, ξ) e^{i x·ξ} ℱ[u](ξ) dξ where p(x, ξ) is a general symbol that may depend on both spatial and frequency variables. It supports both 1D and 2D inputs and includes optional numerical smoothing techniques to enhance stability for non-smooth or oscillatory symbols. Parameters ---------- u_vals : np.ndarray Input function values defined on a uniform spatial grid. Can be 1D (Nx,) or 2D (Nx, Ny). x_grid : np.ndarray Spatial grid points along each axis. In 1D: shape (Nx,). In 2D: tuple of two arrays (X, Y) or list of coordinate arrays. xi_grid : np.ndarray Frequency grid points. In 1D: shape (Nxi,). In 2D: tuple of two arrays (Xi, Eta) or list of frequency arrays. symbol_func : callable A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array. freq_window : {'gaussian', 'hann', None}, optional Type of frequency-domain window to apply for regularization: - 'gaussian': Smooth exponential decay near high frequencies. - 'hann': Cosine-based tapering with hard cutoff. - None: No frequency window applied. clamp : float, optional Maximum absolute value allowed for the symbol to prevent numerical overflow. Default is 1e6. space_window : bool, optional If True, applies a smooth spatial Gaussian window centered in the domain to reduce boundary artifacts. Default is False. Returns ------- np.ndarray The result of applying the pseudo-differential operator Op(p) to u. Shape matches u_vals. Notes ----- - This version does not assume periodicity and is suitable for Dirichlet or Neumann boundary conditions. - In 1D, the integral is evaluated as a sum over (x, ξ), using matrix exponentials. - In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η), which can be computationally intensive. - Symbol evaluation should be vectorized for performance. - For large grids, consider reducing resolution via resampling before calling this function. See Also -------- kohn_nirenberg_fft : Faster implementation for periodic domains using FFT. PseudoDifferentialOperator : Class for symbolic manipulation of pseudo-differential operators. """ if u_vals.ndim == 1: # === 1D case === x = x_grid xi = xi_grid dx = x[1] - x[0] dxi = xi[1] - xi[0] phase_ft = np.exp(-1j * np.outer(xi, x)) # (Nxi, Nx) u_hat = dx * np.dot(phase_ft, u_vals) # (Nxi,) X, XI = np.meshgrid(x, xi, indexing='ij') # (Nx, Nxi) sigma_vals = symbol_func(X, XI) # Clamp values sigma_vals = np.clip(sigma_vals, -clamp, clamp) # Frequency window if freq_window == 'gaussian': sigma = 0.8 * np.max(np.abs(XI)) window = np.exp(-(XI / sigma)**4) sigma_vals *= window elif freq_window == 'hann': window = 0.5 * (1 + np.cos(np.pi * XI / np.max(np.abs(XI)))) sigma_vals *= window * (np.abs(XI) < np.max(np.abs(XI))) # Spatial window if space_window: x_center = (x[0] + x[-1]) / 2 L = (x[-1] - x[0]) / 2 window = np.exp(-((X - x_center)/L)**2) sigma_vals *= window exp_matrix = np.exp(1j * np.outer(x, xi)) # (Nx, Nxi) integrand = sigma_vals * u_hat[np.newaxis, :] * exp_matrix result = dxi * np.sum(integrand, axis=1) / (2 * np.pi) return result elif u_vals.ndim == 2: # === 2D case === x1, x2 = x_grid xi1, xi2 = xi_grid dx1 = x1[1] - x1[0] dx2 = x2[1] - x2[0] dxi1 = xi1[1] - xi1[0] dxi2 = xi2[1] - xi2[0] X1, X2 = np.meshgrid(x1, x2, indexing='ij') XI1, XI2 = np.meshgrid(xi1, xi2, indexing='ij') # Fourier transform of u(x1, x2) phase_ft = np.exp(-1j * (np.tensordot(x1, xi1, axes=0)[:, None, :, None] + np.tensordot(x2, xi2, axes=0)[None, :, None, :])) u_hat = np.tensordot(u_vals, phase_ft, axes=([0,1], [0,1])) * dx1 * dx2 # Symbol evaluation sigma_vals = symbol_func(X1[:, :, None, None], X2[:, :, None, None], XI1[None, None, :, :], XI2[None, None, :, :]) # Clamp values sigma_vals = np.clip(sigma_vals, -clamp, clamp) # Frequency window if freq_window == 'gaussian': sigma_xi1 = 0.8 * np.max(np.abs(XI1)) sigma_xi2 = 0.8 * np.max(np.abs(XI2)) window = np.exp(-(XI1[None, None, :, :] / sigma_xi1)**4 - (XI2[None, None, :, :] / sigma_xi2)**4) sigma_vals *= window elif freq_window == 'hann': # Frequency window - Hanning wx = 0.5 * (1 + np.cos(np.pi * XI1 / np.max(np.abs(XI1)))) wy = 0.5 * (1 + np.cos(np.pi * XI2 / np.max(np.abs(XI2)))) # Mask to zero outside max frequency mask_x = (np.abs(XI1) < np.max(np.abs(XI1))) mask_y = (np.abs(XI2) < np.max(np.abs(XI2))) # Expand wx and wy to match sigma_vals shape: (64, 64, 64, 64) sigma_vals *= wx[:, :, None, None] * wy[:, :, None, None] sigma_vals *= mask_x[:, :, None, None] * mask_y[:, :, None, None] # Spatial window if space_window: x_center = (x1[0] + x1[-1])/2 y_center = (x2[0] + x2[-1])/2 Lx = (x1[-1] - x1[0])/2 Ly = (x2[-1] - x2[0])/2 window = np.exp(-((X1 - x_center)/Lx)**2 - ((X2 - y_center)/Ly)**2) sigma_vals *= window[:, :, None, None] # Oscillatory phase phase = np.exp(1j * (X1[:, :, None, None] * XI1[None, None, :, :] + X2[:, :, None, None] * XI2[None, None, :, :])) integrand = sigma_vals * u_hat[None, None, :, :] * phase result = dxi1 * dxi2 * np.sum(integrand, axis=(2, 3)) / (2 * np.pi)**2 return result else: raise NotImplementedError("Only 1D and 2D supported") def step_ETD_RK4(self, u): """ Perform one Exponential Time Differencing Runge-Kutta of 4th order (ETD-RK4) time step for first-order in time PDEs of the form: ∂ₜu = L u + N(u) where L is a linear operator (possibly nonlocal or pseudo-differential), and N is a nonlinear term treated via pseudo-spectral methods. This method evaluates the exponential integrator up to fourth-order accuracy in time. The ETD-RK4 scheme uses four stages to approximate the integral of the variation-of-constants formula: uⁿ⁺¹ = e^(L Δt) uⁿ + Δt ∫₀¹ e^(L Δt (1 - τ)) φ(N(u(τ))) dτ where φ denotes the nonlinear contributions evaluated at intermediate stages. Parameters u (np.ndarray): Current solution in real space (physical grid values). Returns: np.ndarray: Updated solution in real space after one ETD-RK4 time step. Notes: - The linear part L is diagonal in Fourier space and precomputed as self.L(k). - Nonlinear terms are evaluated in physical space and transformed via FFT. - The functions φ₁(z) and φ₂(z) are entire functions arising from the ETD scheme: φ₁(z) = (eᶻ - 1)/z if z ≠ 0 = 1 if z = 0 φ₂(z) = (eᶻ - 1 - z)/z² if z ≠ 0 = ½ if z = 0 - This implementation assumes periodic boundary conditions and uses spectral differentiation via FFT. - See Hochbruck & Ostermann (2010) for theoretical background on exponential integrators. See Also: step_ETD_RK4_order2 : For second-order in time equations. psiOp_apply : For applying pseudo-differential operators. apply_nonlinear : For handling nonlinear terms in the PDE. """ dt = self.dt L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY) E = np.exp(dt * L_fft) E2 = np.exp(dt * L_fft / 2) def phi1(z): return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1) / z, 1.0) def phi2(z): return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1 - z) / z**2, 0.5) phi1_dtL = phi1(dt * L_fft) phi2_dtL = phi2(dt * L_fft) fft = self.fft ifft = self.ifft u_hat = fft(u) N1 = fft(self.apply_nonlinear(u)) a = ifft(E2 * (u_hat + 0.5 * dt * N1 * phi1_dtL)) N2 = fft(self.apply_nonlinear(a)) b = ifft(E2 * (u_hat + 0.5 * dt * N2 * phi1_dtL)) N3 = fft(self.apply_nonlinear(b)) c = ifft(E * (u_hat + dt * N3 * phi1_dtL)) N4 = fft(self.apply_nonlinear(c)) u_new_hat = E * u_hat + dt * ( N1 * phi1_dtL + 2 * (N2 + N3) * phi2_dtL + N4 * phi1_dtL ) / 6 return ifft(u_new_hat) def step_ETD_RK4_order2(self, u, v): """ Perform one time step of the Exponential Time Differencing Runge-Kutta 4th-order (ETD-RK4) scheme for second-order PDEs. This method evolves the solution u and its time derivative v forward in time by one step using the ETD-RK4 integrator. It is designed for systems of the form: ∂ₜ²u = L u + N(u) where L is a linear operator and N is a nonlinear term computed via self.apply_nonlinear. The exponential integrator handles the linear part exactly in Fourier space, while the nonlinear terms are integrated using a fourth-order Runge-Kutta-like approach. This ensures high accuracy and stability for stiff systems. Parameters: u (np.ndarray): Current solution array in real space. v (np.ndarray): Current time derivative of the solution (∂ₜu) in real space. Returns: tuple: (u_new, v_new), updated solution and its time derivative after one time step. Notes: - Assumes periodic boundary conditions and uses FFT-based spectral methods. - Handles both 1D and 2D problems seamlessly. - Uses phi functions to compute exponential integrators efficiently. - Suitable for wave equations and other second-order evolution equations with stiffness. """ dt = self.dt L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY) fft = self.fft ifft = self.ifft def rhs(u_val): return ifft(L_fft * fft(u_val)) + self.apply_nonlinear(u_val, is_v=False) # Stage A A = rhs(u) ua = u + 0.5 * dt * v va = v + 0.5 * dt * A # Stage B B = rhs(ua) ub = u + 0.5 * dt * va vb = v + 0.5 * dt * B # Stage C C = rhs(ub) uc = u + dt * vb # Stage D D = rhs(uc) # Final update u_new = u + dt * v + (dt**2 / 6.0) * (A + 2*B + 2*C + D) v_new = v + (dt / 6.0) * (A + 2*B + 2*C + D) return u_new, v_new def check_cfl_condition(self): """ Check the CFL (Courant–Friedrichs–Lewymann) condition based on group velocity for second-order time-dependent PDEs. This method verifies whether the chosen time step dt satisfies the numerical stability condition derived from the maximum wave propagation speed in the system. It supports both 1D and 2D problems, with or without a symbolic dispersion relation ω(k). The CFL condition ensures that information does not propagate further than one grid cell per time step. A safety factor of 0.5 is applied by default to ensure robustness. Notes: - In 1D, the group velocity v₉(k) = dω/dk is used to compute the maximum wave speed. - In 2D, the x- and y-directional group velocities are evaluated independently. - If no dispersion relation is available, the imaginary part of the linear operator L(k) is used as an approximation for wave speed. Raises: ------- NotImplementedError: If the spatial dimension is not 1D or 2D. Prints: ------- Warning message if the current time step dt exceeds the CFL-stable limit. """ print("\n*****************") print("* CFL condition *") print("*****************\n") cfl_factor = 0.5 # Safety factor if self.dim == 1: if self.temporal_order == 2 and hasattr(self, 'omega'): k_vals = self.kx omega_vals = np.real(self.omega(k_vals)) with np.errstate(divide='ignore', invalid='ignore'): v_group = np.gradient(omega_vals, k_vals) max_speed = np.max(np.abs(v_group)) else: max_speed = np.max(np.abs(np.imag(self.L(self.kx)))) dx = self.Lx / self.Nx cfl_limit = cfl_factor * dx / max_speed if max_speed != 0 else np.inf if self.dt > cfl_limit: print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}") elif self.dim == 2: if self.temporal_order == 2 and hasattr(self, 'omega'): k_vals = self.kx omega_x = np.real(self.omega(k_vals, 0)) omega_y = np.real(self.omega(0, k_vals)) with np.errstate(divide='ignore', invalid='ignore'): v_group_x = np.gradient(omega_x, k_vals) v_group_y = np.gradient(omega_y, k_vals) max_speed_x = np.max(np.abs(v_group_x)) max_speed_y = np.max(np.abs(v_group_y)) else: max_speed_x = np.max(np.abs(np.imag(self.L(self.kx, 0)))) max_speed_y = np.max(np.abs(np.imag(self.L(0, self.ky)))) dx = self.Lx / self.Nx dy = self.Ly / self.Ny cfl_limit = cfl_factor / (max_speed_x / dx + max_speed_y / dy) if (max_speed_x + max_speed_y) != 0 else np.inf if self.dt > cfl_limit: print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}") else: raise NotImplementedError("Only 1D and 2D problems are supported.") def check_symbol_conditions(self, k_range=None, verbose=True): """ Check strict analytic conditions on the linear symbol self.L_symbolic: This method evaluates three key properties of the Fourier multiplier symbol a(k) = self.L(k), which are crucial for well-posedness, stability, and numerical efficiency. The checks apply to both 1D and 2D cases. Conditions checked: ------------------ 1. **Stability condition**: Re(a(k)) ≤ 0 for all k ≠ 0 Ensures that the system does not exhibit exponential growth in time. 2. **Dissipation condition**: Re(a(k)) ≤ -δ |k|² for large |k| Ensures sufficient damping at high frequencies to avoid oscillatory instability. 3. **Growth condition**: |a(k)| ≤ C (1 + |k|)^m with m ≤ 4 Ensures that the symbol does not grow too rapidly with frequency, which would otherwise cause numerical instability or unphysical amplification. Parameters ---------- k_range : tuple or None, optional Specifies the range of frequencies to test in the form (k_min, k_max, N). If None, defaults are used: [-10, 10] with 500 points in 1D, or [-10, 10] with 100 points per axis in 2D. verbose : bool, default=True If True, prints detailed results of each condition check. Returns: -------- None Output is printed directly to the console for interpretability. Notes: ------ - In 2D, the radial frequency |k| = √(kx² + ky²) is used for comparisons. - The dissipation threshold assumes δ = 0.01 and p = 2 by default. - The growth ratio is compared against |k|⁴; values above 100 indicate rapid growth. - This function is typically called during solver setup or analysis phase. See Also: --------- analyze_wave_propagation : For further symbolic and numerical analysis of dispersion. plot_symbol : Visualizes the symbol's behavior over the frequency domain. """ print("\n********************") print("* Symbol condition *") print("********************\n") if self.dim == 1: if k_range is None: k_vals = np.linspace(-10, 10, 500) else: k_min, k_max, N = k_range k_vals = np.linspace(k_min, k_max, N) L_vals = self.L(k_vals) k_abs = np.abs(k_vals) elif self.dim == 2: if k_range is None: k_vals = np.linspace(-10, 10, 100) else: k_min, k_max, N = k_range k_vals = np.linspace(k_min, k_max, N) KX, KY = np.meshgrid(k_vals, k_vals) L_vals = self.L(KX, KY) k_abs = np.sqrt(KX**2 + KY**2) else: raise ValueError("Only 1D and 2D dimensions are supported.") re_vals = np.real(L_vals) abs_vals = np.abs(L_vals) # === Condition 1: Stability if np.any(re_vals > 1e-12): max_pos = np.max(re_vals) if verbose: print(f"❌ Stability violated: max Re(a(k)) = {max_pos}") print("Unstable symbol: Re(a(k)) > 0") elif verbose: print("✅ Spectral stability satisfied: Re(a(k)) ≤ 0") # === Condition 2: Dissipation mask = k_abs > 2 if np.any(mask): re_decay = re_vals[mask] expected_decay = -0.01 * k_abs[mask]**2 if np.any(re_decay > expected_decay + 1e-6): if verbose: print("⚠️ Insufficient high-frequency dissipation") else: if verbose: print("✅ Proper high-frequency dissipation") # === Condition 3: Growth growth_ratio = abs_vals / (1 + k_abs)**4 if np.max(growth_ratio) > 100: if verbose: print("⚠️ Symbol grows rapidly: |a(k)| ≳ |k|^4") else: if verbose: print("✅ Reasonable spectral growth") if verbose: print("✔ Symbol analysis completed.") def analyze_wave_propagation(self): """ Perform a detailed analysis of wave propagation characteristics based on the dispersion relation ω(k). This method visualizes key wave properties in both 1D and 2D settings: - Dispersion relation: ω(k) - Phase velocity: v_p(k) = ω(k)/|k| - Group velocity: v_g(k) = ∇ₖ ω(k) - Anisotropy in 2D (via magnitude of group velocity) The symbolic dispersion relation 'omega_symbolic' must be defined beforehand. This is typically available only for second-order-in-time equations. In 1D: Plots ω(k), v_p(k), and v_g(k) over a range of k values. In 2D: Displays heatmaps of ω(kx, ky), v_p(kx, ky), and |v_g(kx, ky)| over a 2D wavenumber grid. Raises: AttributeError: If 'omega_symbolic' is not defined, the method exits gracefully with a message. Side Effects: Generates and displays matplotlib plots. """ print("\n*****************************") print("* Wave propagation analysis *") print("*****************************\n") if not hasattr(self, 'omega_symbolic'): print("❌ omega_symbolic not defined. Only available for 2nd order in time.") return if self.dim == 1: k = self.k_symbols[0] omega_func = lambdify(k, self.omega_symbolic, 'numpy') k_vals = np.linspace(-10, 10, 1000) omega_vals = omega_func(k_vals) with np.errstate(divide='ignore', invalid='ignore'): v_phase = np.where(k_vals != 0, omega_vals / k_vals, 0.0) dk = k_vals[1] - k_vals[0] v_group = np.gradient(omega_vals, dk) plt.figure(figsize=(10, 6)) plt.plot(k_vals, omega_vals, label=r'$\omega(k)$') plt.plot(k_vals, v_phase, label=r'$v_p(k)$') plt.plot(k_vals, v_group, label=r'$v_g(k)$') plt.title("1D Wave Propagation Analysis") plt.xlabel("k") plt.grid() plt.legend() plt.tight_layout() plt.show() elif self.dim == 2: kx, ky = self.k_symbols omega_func = lambdify((kx, ky), self.omega_symbolic, 'numpy') k_vals = np.linspace(-10, 10, 200) KX, KY = np.meshgrid(k_vals, k_vals) K_mag = np.sqrt(KX**2 + KY**2) K_mag[K_mag == 0] = 1e-8 # Avoid division by 0 omega_vals = omega_func(KX, KY) v_phase = np.real(omega_vals) / K_mag dk = k_vals[1] - k_vals[0] domega_dx = np.gradient(omega_vals, dk, axis=0) domega_dy = np.gradient(omega_vals, dk, axis=1) v_group_norm = np.sqrt(np.abs(domega_dx)**2 + np.abs(domega_dy)**2) fig, axs = plt.subplots(1, 3, figsize=(18, 5)) im0 = axs[0].imshow(np.real(omega_vals), extent=[-10, 10, -10, 10], origin='lower', cmap='viridis') axs[0].set_title(r'$\omega(k_x, k_y)$') plt.colorbar(im0, ax=axs[0]) im1 = axs[1].imshow(v_phase, extent=[-10, 10, -10, 10], origin='lower', cmap='plasma') axs[1].set_title(r'$v_p(k_x, k_y)$') plt.colorbar(im1, ax=axs[1]) im2 = axs[2].imshow(v_group_norm, extent=[-10, 10, -10, 10], origin='lower', cmap='inferno') axs[2].set_title(r'$|v_g(k_x, k_y)|$') plt.colorbar(im2, ax=axs[2]) for ax in axs: ax.set_xlabel(r'$k_x$') ax.set_ylabel(r'$k_y$') ax.set_aspect('equal') plt.tight_layout() plt.show() else: print("❌ Only 1D and 2D wave analysis supported.") def plot_symbol(self, component="abs", k_range=None, cmap="viridis"): """ Visualize the spectral symbol L(k) or L(kx, ky) in 1D or 2D. This method plots the linear operator's symbolic Fourier representation either as a function of a single wavenumber k (1D), or two wavenumbers kx and ky (2D). The user can choose to display the real part, imaginary part, or absolute value of the symbol. Parameters ---------- component : str {'abs', 're', 'im'} Component of the symbol to visualize: - 'abs' : absolute value |a(k)| - 're' : real part Re[a(k)] - 'im' : imaginary part Im[a(k)] k_range : tuple (kmin, kmax, N), optional Wavenumber range for evaluation: - kmin: minimum wavenumber - kmax: maximum wavenumber - N: number of sampling points If None, defaults to [-10, 10] with high resolution. cmap : str, optional Colormap used for 2D surface plots. Default is 'viridis'. Raises ------ ValueError: If the spatial dimension is not 1D or 2D. Notes: - In 1D, the symbol is plotted using a standard 2D line plot. - In 2D, a 3D surface plot is generated with color-mapped height. - Symbol evaluation uses self.L(k), which must be defined and callable. """ print("\n*******************") print("* Symbol plotting *") print("*******************\n") assert component in ("abs", "re", "im"), "component must be 'abs', 're' or 'im'" if self.dim == 1: if k_range is None: k_vals = np.linspace(-10, 10, 1000) else: kmin, kmax, N = k_range k_vals = np.linspace(kmin, kmax, N) L_vals = self.L(k_vals) if component == "re": vals = np.real(L_vals) label = "Re[a(k)]" elif component == "im": vals = np.imag(L_vals) label = "Im[a(k)]" else: vals = np.abs(L_vals) label = "|a(k)|" plt.plot(k_vals, vals) plt.xlabel("k") plt.ylabel(label) plt.title(f"Spectral symbol: {label}") plt.grid(True) plt.show() elif self.dim == 2: if k_range is None: k_vals = np.linspace(-10, 10, 300) else: kmin, kmax, N = k_range k_vals = np.linspace(kmin, kmax, N) KX, KY = np.meshgrid(k_vals, k_vals) L_vals = self.L(KX, KY) if component == "re": Z = np.real(L_vals) title = "Re[a(kx, ky)]" elif component == "im": Z = np.imag(L_vals) title = "Im[a(kx, ky)]" else: Z = np.abs(L_vals) title = "|a(kx, ky)|" fig = plt.figure(figsize=(8, 6)) ax = fig.add_subplot(111, projection='3d') surf = ax.plot_surface(KX, KY, Z, cmap=cmap, edgecolor='none', antialiased=True) fig.colorbar(surf, ax=ax, shrink=0.6) ax.set_xlabel("kx") ax.set_ylabel("ky") ax.set_zlabel(title) ax.set_title(f"2D spectral symbol: {title}") plt.tight_layout() plt.show() else: raise ValueError("Only 1D and 2D supported.") def compute_energy(self): """ Compute the total energy of the wave equation solution for second-order temporal PDEs. The energy is defined as: E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹ᐟ²u|² ] dx where L is the linear operator associated with the spatial part of the PDE, and L¹ᐟ² denotes its square root in Fourier space. This method supports both 1D and 2D problems and is only meaningful when self.temporal_order == 2 (second-order time derivative). Returns ------- float or None: Total energy at current time step. Returns None if the temporal order is not 2 or if no valid velocity data (v_prev) is available. Notes ----- - Uses FFT-based spectral differentiation to compute the spatial contributions. - Assumes periodic boundary conditions. - Handles both real and complex-valued solutions. """ if self.temporal_order != 2 or self.v_prev is None: return None u = self.u_prev v = self.v_prev # Fourier transform of u u_hat = self.fft(u) if self.dim == 1: # 1D case L_vals = self.L(self.KX) sqrt_L = np.sqrt(np.abs(L_vals)) Lu_hat = sqrt_L * u_hat # Apply sqrt(|L(k)|) in Fourier space Lu = self.ifft(Lu_hat) dx = self.Lx / self.Nx energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2) total_energy = np.sum(energy_density) * dx elif self.dim == 2: # 2D case L_vals = self.L(self.KX, self.KY) sqrt_L = np.sqrt(np.abs(L_vals)) Lu_hat = sqrt_L * u_hat Lu = self.ifft(Lu_hat) dx = self.Lx / self.Nx dy = self.Ly / self.Ny energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2) total_energy = np.sum(energy_density) * dx * dy else: raise ValueError("Unsupported dimension for u.") return total_energy def plot_energy(self, log=False): """ Plot the time evolution of the total energy for wave equations. Visualizes the energy computed during simulation for both 1D and 2D cases. Requires temporal_order=2 and prior execution of compute_energy() during solve(). Parameters: log : bool If True, displays energy on a logarithmic scale to highlight exponential decay/growth. Notes: - Energy is defined as E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹⸍²u|² ] dx - Only available if energy monitoring was activated in solve() - Automatically skips plotting if no energy data is available Displays: - Time vs. Total Energy plot with grid and legend - Appropriate axis labels and dimensional context (1D/2D) - Logarithmic or linear scaling based on input parameter """ if not hasattr(self, 'energy_history') or not self.energy_history: print("No energy data recorded. Call compute_energy() within solve().") return # Time vector for plotting t = np.linspace(0, self.Lt, len(self.energy_history)) # Create the figure plt.figure(figsize=(6, 4)) if log: plt.semilogy(t, self.energy_history, label="Energy (log scale)") else: plt.plot(t, self.energy_history, label="Energy") # Axis labels and title plt.xlabel("Time") plt.ylabel("Total energy") plt.title("Energy evolution ({}D)".format(self.dim)) # Display options plt.grid(True) plt.legend() plt.tight_layout() plt.show() def show_stationary_solution(self, u=None, component='abs', cmap='viridis'): """ Display the stationary solution computed by solve_stationary_psiOp. This method visualizes the solution of a pseudo-differential equation solved in stationary mode. It supports both 1D and 2D spatial domains, with options to display different components of the solution (real, imaginary, absolute value, or phase). Parameters ---------- u : ndarray, optional Precomputed solution array. If None, calls solve_stationary_psiOp() to compute the solution. component : str, optional {'real', 'imag', 'abs', 'angle'} Component of the complex-valued solution to display: - 'real': Real part - 'imag': Imaginary part - 'abs' : Absolute value (modulus) - 'angle' : Phase (argument) cmap : str, optional Colormap used for 2D visualization (default: 'viridis'). Raises ------ ValueError If an invalid component is specified or if the spatial dimension is not supported (only 1D and 2D are implemented). Notes ----- - In 1D, the solution is displayed using a standard line plot. - In 2D, the solution is visualized as a 3D surface plot. """ def get_component(u): if component == 'real': return np.real(u) elif component == 'imag': return np.imag(u) elif component == 'abs': return np.abs(u) elif component == 'angle': return np.angle(u) else: raise ValueError("Invalid component") if u is None: u = self.solve_stationary_psiOp() if self.dim == 1: # Plot the solution in 1D plt.figure(figsize=(8, 4)) plt.plot(self.x_grid, get_component(u), label=f'{component} of u') plt.xlabel('x') plt.ylabel(f'{component} of u') plt.title('Stationary solution (1D)') plt.grid(True) plt.legend() plt.tight_layout() plt.show() elif self.dim == 2: fig = plt.figure(figsize=(12, 6)) ax = fig.add_subplot(111, projection='3d') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') plt.title('Stationary solution (2D)') data0 = get_component(u) ax.plot_surface(self.X, self.Y, data0, cmap='viridis') plt.tight_layout() plt.show() else: raise ValueError("Only 1D and 2D display are supported.") def animate(self, component='abs', overlay='contour'): """ Create an animated plot of the solution evolution over time. This method generates a dynamic visualization of the solution array `self.frames`, animating either the real part, imaginary part, absolute value, or complex angle of the field. It supports both 1D line plots and 2D surface plots with optional contour overlays. Parameters ---------- component : str in {'real', 'imag', 'abs', 'angle'} The component of the solution to visualize: - 'real' : Real part Re(u) - 'imag' : Imaginary part Im(u) - 'abs' : Absolute value |u| - 'angle' : Complex argument arg(u) overlay : str in {'contour', 'front'}, optional Type of overlay for 2D animations: - 'contour' : Adds contour lines beneath the surface at each frame. - 'front' : Used for tracking wavefronts. Returns ------- FuncAnimation A Matplotlib `FuncAnimation` object that can be displayed or saved as a video. Notes ----- - Uses linear interpolation to map simulation frames to target animation frames. - In 2D, the z-axis dynamically rescales based on current data range. - For 'angle' component, color scaling is fixed between -π and π for consistency. - The animation interval is fixed at 50 ms per frame for smooth playback. """ def get_component(u): if component == 'real': return np.real(u) elif component == 'imag': return np.imag(u) elif component == 'abs': return np.abs(u) elif component == 'angle': return np.angle(u) else: raise ValueError("Invalid component") print("\n*********************") print("* Solution plotting *") print("*********************\n") # === Calculate time vector of stored frames === save_interval = max(1, self.Nt // self.n_frames) frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt) # === Target times for animation === target_times = np.linspace(0, self.Lt, self.n_frames // 2) # Map target times to nearest frame indices frame_indices = [np.argmin(np.abs(frame_times - t)) for t in target_times] if self.dim == 1: fig, ax = plt.subplots() line, = ax.plot(self.X, get_component(self.frames[0])) ax.set_ylim(np.min(self.frames[0]), np.max(self.frames[0])) ax.set_xlabel('x') ax.set_ylabel(f'{component} of u') ax.set_title('Initial condition') plt.tight_layout() plt.show() def update(frame_number): frame = frame_indices[frame_number] ydata = get_component(self.frames[frame]) ydata_real = np.real(ydata) if np.iscomplexobj(ydata) else ydata line.set_ydata(ydata_real) ax.set_ylim(np.min(ydata_real), np.max(ydata_real)) current_time = target_times[frame_number] ax.set_title(f't = {current_time:.2f}') return line, ani = FuncAnimation(fig, update, frames=len(target_times), interval=50) return ani else: # dim == 2 fig = plt.figure(figsize=(14, 8)) ax = fig.add_subplot(111, projection='3d') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') ax.zaxis.labelpad = 0 ax.set_title('Initial condition') data0 = get_component(self.frames[0]) surf = [ax.plot_surface(self.X, self.Y, data0, cmap='viridis')] plt.show() def update(frame_number): frame = frame_indices[frame_number] current_data = get_component(self.frames[frame]) z_offset = np.max(current_data) + 0.05 * (np.max(current_data) - np.min(current_data)) ax.clear() surf[0] = ax.plot_surface(self.X, self.Y, current_data, cmap='viridis', vmin=-1, vmax=1 if component != 'angle' else np.pi) if overlay == 'contour': ax.contour(self.X, self.Y, current_data, levels=10, cmap='cool', offset=z_offset) elif overlay == 'front': dx = self.x_grid[1] - self.x_grid[0] dy = self.y_grid[1] - self.y_grid[0] du_dx, du_dy = np.gradient(current_data, dx, dy) grad_norm = np.sqrt(du_dx**2 + du_dy**2) local_max = (grad_norm == maximum_filter(grad_norm, size=5)) normalized = grad_norm[local_max] / np.max(grad_norm) colors = cm.plasma(normalized) ax.scatter(self.X[local_max], self.Y[local_max], z_offset * np.ones_like(self.X[local_max]), color=colors, s=10, alpha=0.8) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') current_time = target_times[frame_number] ax.set_title(f'Solution at t = {current_time:.2f}') return surf ani = FuncAnimation(fig, update, frames=len(target_times), interval=50) return ani def test(self, u_exact, t_eval=None, norm='relative', threshold=1e-2, plot=True, component='real'): """ Test the solver against an exact solution. This method quantitatively compares the numerical solution with a provided exact solution at a specified time using either relative or absolute error norms. It supports both stationary and time-dependent problems in 1D and 2D. If enabled, it also generates plots of the solution, exact solution, and pointwise error. Parameters ---------- u_exact : callable Exact solution function taking spatial coordinates and optionally time as arguments. t_eval : float, optional Time at which to compare solutions. For non-stationary problems, defaults to final time Lt. Ignored for stationary problems. norm : str {'relative', 'absolute'} Type of error norm used in comparison. threshold : float Acceptable error threshold; raises an assertion if exceeded. plot : bool Whether to display visual comparison plots (default: True). component : str {'real', 'imag', 'abs'} Component of the solution to compare and visualize. Raises ------ ValueError If unsupported dimension is encountered or requested evaluation time exceeds simulation duration. AssertionError If computed error exceeds the given threshold. Prints ------ - Information about the closest available frame to the requested evaluation time. - Computed error value and comparison to threshold. Notes ----- - For time-dependent problems, the solution is extracted from precomputed frames. - Plots are adapted to spatial dimension: line plots for 1D, image plots for 2D. - The method ensures consistent handling of real, imaginary, and magnitude components. """ if self.is_stationary: print("Testing a stationary solution.") u_num = self.u # Compute exact solution if self.dim == 1: u_ex = u_exact(self.X) elif self.dim == 2: u_ex = u_exact(self.X, self.Y) else: raise ValueError("Unsupported dimension.") actual_t = None else: if t_eval is None: t_eval = self.Lt save_interval = max(1, self.Nt // self.n_frames) frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt) frame_index = np.argmin(np.abs(frame_times - t_eval)) actual_t = frame_times[frame_index] print(f"Closest available time to t_eval={t_eval}: {actual_t}") if frame_index >= len(self.frames): raise ValueError(f"Time t = {t_eval} exceeds simulation duration.") u_num = self.frames[frame_index] # Compute exact solution at the actual time if self.dim == 1: u_ex = u_exact(self.X, actual_t) elif self.dim == 2: u_ex = u_exact(self.X, self.Y, actual_t) else: raise ValueError("Unsupported dimension.") # Select component if component == 'real': diff = np.real(u_num) - np.real(u_ex) ref = np.real(u_ex) elif component == 'imag': diff = np.imag(u_num) - np.imag(u_ex) ref = np.imag(u_ex) elif component == 'abs': diff = np.abs(u_num) - np.abs(u_ex) ref = np.abs(u_ex) else: raise ValueError("Invalid component.") # Compute error if norm == 'relative': error = np.linalg.norm(diff) / np.linalg.norm(ref) elif norm == 'absolute': error = np.linalg.norm(diff) else: raise ValueError("Unknown norm type.") label_time = f"t = {actual_t}" if actual_t is not None else "" print(f"Test error {label_time}: {error:.3e}") assert error < threshold, f"Error too large {label_time}: {error:.3e}" # Plot if plot: if self.dim == 1: plt.figure(figsize=(12, 6)) plt.subplot(2, 1, 1) plt.plot(self.X, np.real(u_num), label='Numerical') plt.plot(self.X, np.real(u_ex), '--', label='Exact') plt.title(f'Solution {label_time}, error = {error:.2e}') plt.legend() plt.grid() plt.subplot(2, 1, 2) plt.plot(self.X, np.abs(diff), color='red') plt.title('Absolute Error') plt.grid() plt.tight_layout() plt.show() else: plt.figure(figsize=(15, 5)) plt.subplot(1, 3, 1) plt.title("Numerical Solution") plt.imshow(np.abs(u_num), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis') plt.colorbar() plt.subplot(1, 3, 2) plt.title("Exact Solution") plt.imshow(np.abs(u_ex), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis') plt.colorbar() plt.subplot(1, 3, 3) plt.title(f"Error (Norm = {error:.2e})") plt.imshow(np.abs(diff), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='inferno') plt.colorbar() plt.tight_layout() plt.show()Methods
def analyze_wave_propagation(self)-
Perform a detailed analysis of wave propagation characteristics based on the dispersion relation ω(k).
This method visualizes key wave properties in both 1D and 2D settings:
- Dispersion relation: ω(k)
- Phase velocity: v_p(k) = ω(k)/|k|
- Group velocity: v_g(k) = ∇ₖ ω(k)
- Anisotropy in 2D (via magnitude of group velocity)
The symbolic dispersion relation 'omega_symbolic' must be defined beforehand. This is typically available only for second-order-in-time equations.
In 1D: Plots ω(k), v_p(k), and v_g(k) over a range of k values.
In 2D: Displays heatmaps of ω(kx, ky), v_p(kx, ky), and |v_g(kx, ky)| over a 2D wavenumber grid.
Raises
AttributeError- If 'omega_symbolic' is not defined, the method exits gracefully with a message.
Side Effects: Generates and displays matplotlib plots.
Expand source code
def analyze_wave_propagation(self): """ Perform a detailed analysis of wave propagation characteristics based on the dispersion relation ω(k). This method visualizes key wave properties in both 1D and 2D settings: - Dispersion relation: ω(k) - Phase velocity: v_p(k) = ω(k)/|k| - Group velocity: v_g(k) = ∇ₖ ω(k) - Anisotropy in 2D (via magnitude of group velocity) The symbolic dispersion relation 'omega_symbolic' must be defined beforehand. This is typically available only for second-order-in-time equations. In 1D: Plots ω(k), v_p(k), and v_g(k) over a range of k values. In 2D: Displays heatmaps of ω(kx, ky), v_p(kx, ky), and |v_g(kx, ky)| over a 2D wavenumber grid. Raises: AttributeError: If 'omega_symbolic' is not defined, the method exits gracefully with a message. Side Effects: Generates and displays matplotlib plots. """ print("\n*****************************") print("* Wave propagation analysis *") print("*****************************\n") if not hasattr(self, 'omega_symbolic'): print("❌ omega_symbolic not defined. Only available for 2nd order in time.") return if self.dim == 1: k = self.k_symbols[0] omega_func = lambdify(k, self.omega_symbolic, 'numpy') k_vals = np.linspace(-10, 10, 1000) omega_vals = omega_func(k_vals) with np.errstate(divide='ignore', invalid='ignore'): v_phase = np.where(k_vals != 0, omega_vals / k_vals, 0.0) dk = k_vals[1] - k_vals[0] v_group = np.gradient(omega_vals, dk) plt.figure(figsize=(10, 6)) plt.plot(k_vals, omega_vals, label=r'$\omega(k)$') plt.plot(k_vals, v_phase, label=r'$v_p(k)$') plt.plot(k_vals, v_group, label=r'$v_g(k)$') plt.title("1D Wave Propagation Analysis") plt.xlabel("k") plt.grid() plt.legend() plt.tight_layout() plt.show() elif self.dim == 2: kx, ky = self.k_symbols omega_func = lambdify((kx, ky), self.omega_symbolic, 'numpy') k_vals = np.linspace(-10, 10, 200) KX, KY = np.meshgrid(k_vals, k_vals) K_mag = np.sqrt(KX**2 + KY**2) K_mag[K_mag == 0] = 1e-8 # Avoid division by 0 omega_vals = omega_func(KX, KY) v_phase = np.real(omega_vals) / K_mag dk = k_vals[1] - k_vals[0] domega_dx = np.gradient(omega_vals, dk, axis=0) domega_dy = np.gradient(omega_vals, dk, axis=1) v_group_norm = np.sqrt(np.abs(domega_dx)**2 + np.abs(domega_dy)**2) fig, axs = plt.subplots(1, 3, figsize=(18, 5)) im0 = axs[0].imshow(np.real(omega_vals), extent=[-10, 10, -10, 10], origin='lower', cmap='viridis') axs[0].set_title(r'$\omega(k_x, k_y)$') plt.colorbar(im0, ax=axs[0]) im1 = axs[1].imshow(v_phase, extent=[-10, 10, -10, 10], origin='lower', cmap='plasma') axs[1].set_title(r'$v_p(k_x, k_y)$') plt.colorbar(im1, ax=axs[1]) im2 = axs[2].imshow(v_group_norm, extent=[-10, 10, -10, 10], origin='lower', cmap='inferno') axs[2].set_title(r'$|v_g(k_x, k_y)|$') plt.colorbar(im2, ax=axs[2]) for ax in axs: ax.set_xlabel(r'$k_x$') ax.set_ylabel(r'$k_y$') ax.set_aspect('equal') plt.tight_layout() plt.show() else: print("❌ Only 1D and 2D wave analysis supported.") def animate(self, component='abs', overlay='contour')-
Create an animated plot of the solution evolution over time.
This method generates a dynamic visualization of the solution array
self.frames, animating either the real part, imaginary part, absolute value, or complex angle of the field. It supports both 1D line plots and 2D surface plots with optional contour overlays.Parameters
component:str in {'real', 'imag', 'abs', 'angle'}-
The component of the solution to visualize:
- 'real' : Real part Re(u)
- 'imag' : Imaginary part Im(u)
- 'abs' : Absolute value |u|
- 'angle' : Complex argument arg(u)
overlay:str in {'contour', 'front'}, optional-
Type of overlay for 2D animations:
- 'contour' : Adds contour lines beneath the surface at each frame.
- 'front' : Used for tracking wavefronts.
Returns
FuncAnimation- A Matplotlib
FuncAnimationobject that can be displayed or saved as a video.
Notes
- Uses linear interpolation to map simulation frames to target animation frames.
- In 2D, the z-axis dynamically rescales based on current data range.
- For 'angle' component, color scaling is fixed between -π and π for consistency.
- The animation interval is fixed at 50 ms per frame for smooth playback.
Expand source code
def animate(self, component='abs', overlay='contour'): """ Create an animated plot of the solution evolution over time. This method generates a dynamic visualization of the solution array `self.frames`, animating either the real part, imaginary part, absolute value, or complex angle of the field. It supports both 1D line plots and 2D surface plots with optional contour overlays. Parameters ---------- component : str in {'real', 'imag', 'abs', 'angle'} The component of the solution to visualize: - 'real' : Real part Re(u) - 'imag' : Imaginary part Im(u) - 'abs' : Absolute value |u| - 'angle' : Complex argument arg(u) overlay : str in {'contour', 'front'}, optional Type of overlay for 2D animations: - 'contour' : Adds contour lines beneath the surface at each frame. - 'front' : Used for tracking wavefronts. Returns ------- FuncAnimation A Matplotlib `FuncAnimation` object that can be displayed or saved as a video. Notes ----- - Uses linear interpolation to map simulation frames to target animation frames. - In 2D, the z-axis dynamically rescales based on current data range. - For 'angle' component, color scaling is fixed between -π and π for consistency. - The animation interval is fixed at 50 ms per frame for smooth playback. """ def get_component(u): if component == 'real': return np.real(u) elif component == 'imag': return np.imag(u) elif component == 'abs': return np.abs(u) elif component == 'angle': return np.angle(u) else: raise ValueError("Invalid component") print("\n*********************") print("* Solution plotting *") print("*********************\n") # === Calculate time vector of stored frames === save_interval = max(1, self.Nt // self.n_frames) frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt) # === Target times for animation === target_times = np.linspace(0, self.Lt, self.n_frames // 2) # Map target times to nearest frame indices frame_indices = [np.argmin(np.abs(frame_times - t)) for t in target_times] if self.dim == 1: fig, ax = plt.subplots() line, = ax.plot(self.X, get_component(self.frames[0])) ax.set_ylim(np.min(self.frames[0]), np.max(self.frames[0])) ax.set_xlabel('x') ax.set_ylabel(f'{component} of u') ax.set_title('Initial condition') plt.tight_layout() plt.show() def update(frame_number): frame = frame_indices[frame_number] ydata = get_component(self.frames[frame]) ydata_real = np.real(ydata) if np.iscomplexobj(ydata) else ydata line.set_ydata(ydata_real) ax.set_ylim(np.min(ydata_real), np.max(ydata_real)) current_time = target_times[frame_number] ax.set_title(f't = {current_time:.2f}') return line, ani = FuncAnimation(fig, update, frames=len(target_times), interval=50) return ani else: # dim == 2 fig = plt.figure(figsize=(14, 8)) ax = fig.add_subplot(111, projection='3d') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') ax.zaxis.labelpad = 0 ax.set_title('Initial condition') data0 = get_component(self.frames[0]) surf = [ax.plot_surface(self.X, self.Y, data0, cmap='viridis')] plt.show() def update(frame_number): frame = frame_indices[frame_number] current_data = get_component(self.frames[frame]) z_offset = np.max(current_data) + 0.05 * (np.max(current_data) - np.min(current_data)) ax.clear() surf[0] = ax.plot_surface(self.X, self.Y, current_data, cmap='viridis', vmin=-1, vmax=1 if component != 'angle' else np.pi) if overlay == 'contour': ax.contour(self.X, self.Y, current_data, levels=10, cmap='cool', offset=z_offset) elif overlay == 'front': dx = self.x_grid[1] - self.x_grid[0] dy = self.y_grid[1] - self.y_grid[0] du_dx, du_dy = np.gradient(current_data, dx, dy) grad_norm = np.sqrt(du_dx**2 + du_dy**2) local_max = (grad_norm == maximum_filter(grad_norm, size=5)) normalized = grad_norm[local_max] / np.max(grad_norm) colors = cm.plasma(normalized) ax.scatter(self.X[local_max], self.Y[local_max], z_offset * np.ones_like(self.X[local_max]), color=colors, s=10, alpha=0.8) ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') current_time = target_times[frame_number] ax.set_title(f'Solution at t = {current_time:.2f}') return surf ani = FuncAnimation(fig, update, frames=len(target_times), interval=50) return ani def apply_boundary(self, u)-
Apply boundary conditions to the solution array based on the specified type.
This method supports two types of boundary conditions:
- 'periodic': Enforces periodicity by copying opposite boundary values.
- 'dirichlet': Sets all boundary values to zero (homogeneous Dirichlet condition).
Parameters
u:np.ndarray- The solution array representing the field values on a spatial grid. In 1D, shape must be (Nx,). In 2D, shape must be (Nx, Ny).
Raises
ValueError- If
self.boundary_conditionis not one of {'periodic', 'dirichlet'}.
Notes
- For 'periodic':
- In 1D: u[0] = u[-2], u[-1] = u[1]
- In 2D: First and last rows/columns are set equal to their neighbors.
- For 'dirichlet':
- All boundary points are explicitly set to zero.
Expand source code
def apply_boundary(self, u): """ Apply boundary conditions to the solution array based on the specified type. This method supports two types of boundary conditions: - 'periodic': Enforces periodicity by copying opposite boundary values. - 'dirichlet': Sets all boundary values to zero (homogeneous Dirichlet condition). Parameters ---------- u : np.ndarray The solution array representing the field values on a spatial grid. In 1D, shape must be (Nx,). In 2D, shape must be (Nx, Ny). Raises ------ ValueError If `self.boundary_condition` is not one of {'periodic', 'dirichlet'}. Notes ----- - For 'periodic': * In 1D: u[0] = u[-2], u[-1] = u[1] * In 2D: First and last rows/columns are set equal to their neighbors. - For 'dirichlet': * All boundary points are explicitly set to zero. """ if self.boundary_condition == 'periodic': if self.dim == 1: u[0] = u[-2] u[-1] = u[1] elif self.dim == 2: u[0, :] = u[-2, :] u[-1, :] = u[1, :] u[:, 0] = u[:, -2] u[:, -1] = u[:, 1] elif self.boundary_condition == 'dirichlet': if self.dim == 1: u[0] = 0 u[-1] = 0 elif self.dim == 2: u[0, :] = 0 u[-1, :] = 0 u[:, 0] = 0 u[:, -1] = 0 else: raise ValueError( f"Invalid boundary condition '{self.boundary_condition}'. " "Supported types are 'periodic' and 'dirichlet'." ) def apply_nonlinear(self, u, is_v=False)-
Apply nonlinear terms to the solution using spectral differentiation with dealiasing.
This method evaluates all nonlinear terms present in the PDE by substituting spatial derivatives with their spectral approximations computed via FFT. The dealiasing mask ensures numerical stability by removing high-frequency components that could lead to aliasing errors.
Parameters
u:numpy.ndarray- Current solution array on the spatial grid.
is_v:bool- If True, evaluates nonlinear terms for the velocity field v instead of u.
Returns
numpy.ndarray- Array representing the contribution of nonlinear terms multiplied by dt.
Notes:
- In 1D, computes ∂ₓu via FFT and substitutes any derivative term in the nonlinear expressions.
- In 2D, computes ∂ₓu and ∂ᵧu via FFT and performs similar substitutions.
- Uses lambdify to evaluate symbolic nonlinear expressions numerically.
- Derivatives are replaced symbolically with 'u_x' and 'u_y' before evaluation.
Expand source code
def apply_nonlinear(self, u, is_v=False): """ Apply nonlinear terms to the solution using spectral differentiation with dealiasing. This method evaluates all nonlinear terms present in the PDE by substituting spatial derivatives with their spectral approximations computed via FFT. The dealiasing mask ensures numerical stability by removing high-frequency components that could lead to aliasing errors. Parameters ---------- u : numpy.ndarray Current solution array on the spatial grid. is_v : bool If True, evaluates nonlinear terms for the velocity field v instead of u. Returns: numpy.ndarray: Array representing the contribution of nonlinear terms multiplied by dt. Notes: - In 1D, computes ∂ₓu via FFT and substitutes any derivative term in the nonlinear expressions. - In 2D, computes ∂ₓu and ∂ᵧu via FFT and performs similar substitutions. - Uses lambdify to evaluate symbolic nonlinear expressions numerically. - Derivatives are replaced symbolically with 'u_x' and 'u_y' before evaluation. """ if not self.nonlinear_terms: return np.zeros_like(u, dtype=np.complex128) nonlinear_term = np.zeros_like(u, dtype=np.complex128) if self.dim == 1: u_hat = self.fft(u) u_hat *= self.dealiasing_mask u = self.ifft(u_hat) u_x_hat = (1j * self.KX) * u_hat u_x = self.ifft(u_x_hat) for term in self.nonlinear_terms: term_replaced = term if term.has(Derivative): for deriv in term.atoms(Derivative): if deriv.args[1][0] == self.x: term_replaced = term_replaced.subs(deriv, symbols('u_x')) term_func = lambdify((self.t, self.x, self.u_eq, 'u_x'), term_replaced, 'numpy') if is_v: nonlinear_term += term_func(0, self.X, self.v_prev, u_x) else: nonlinear_term += term_func(0, self.X, u, u_x) elif self.dim == 2: u_hat = self.fft(u) u_hat *= self.dealiasing_mask u = self.ifft(u_hat) u_x_hat = (1j * self.KX) * u_hat u_y_hat = (1j * self.KY) * u_hat u_x = self.ifft(u_x_hat) u_y = self.ifft(u_y_hat) for term in self.nonlinear_terms: term_replaced = term if term.has(Derivative): for deriv in term.atoms(Derivative): if deriv.args[1][0] == self.x: term_replaced = term_replaced.subs(deriv, symbols('u_x')) elif deriv.args[1][0] == self.y: term_replaced = term_replaced.subs(deriv, symbols('u_y')) term_func = lambdify((self.t, self.x, self.y, self.u_eq, 'u_x', 'u_y'), term_replaced, 'numpy') if is_v: nonlinear_term += term_func(0, self.X, self.Y, self.v_prev, u_x, u_y) else: nonlinear_term += term_func(0, self.X, self.Y, u, u_x, u_y) else: raise ValueError("Unsupported spatial dimension.") return nonlinear_term * self.dt def apply_psiOp(self, u)-
Apply the pseudo-differential operator to the input field u.
This method dispatches the application of the pseudo-differential operator based on:
- Whether the symbol is spatially dependent (x/y)
- The boundary condition in use (periodic or dirichlet)
Supported operations:
- Constant-coefficient symbols: applied via Fourier multiplication.
- Spatially varying symbols: applied via Kohn–Nirenberg quantization.
- Dirichlet boundary conditions: handled with non-periodic convolution-like quantization.
Dispatch Logic:
if not self.is_spatial: u ↦ Op(p)(D) ⋅ u = 𝓕⁻¹[ p(ξ) ⋅ 𝓕(u) ]
elif periodic: u ↦ Op(p)(x,D) ⋅ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ based of FFT (quicker)
elif dirichlet: u ↦ Op(p)(x,D) ⋅ u ≈ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ (slower)
Parameters
u:np.ndarray- Input field to which the operator is applied. Should be 1D or 2D depending on the problem dimension.
Returns
np.ndarray- Result of applying the pseudo-differential operator to u.
Raises
ValueError- If an unsupported boundary condition is specified.
Expand source code
def apply_psiOp(self, u): """ Apply the pseudo-differential operator to the input field u. This method dispatches the application of the pseudo-differential operator based on: - Whether the symbol is spatially dependent (x/y) - The boundary condition in use (periodic or dirichlet) Supported operations: - Constant-coefficient symbols: applied via Fourier multiplication. - Spatially varying symbols: applied via Kohn–Nirenberg quantization. - Dirichlet boundary conditions: handled with non-periodic convolution-like quantization. Dispatch Logic:\n if not self.is_spatial: u ↦ Op(p)(D) ⋅ u = 𝓕⁻¹[ p(ξ) ⋅ 𝓕(u) ]\n elif periodic: u ↦ Op(p)(x,D) ⋅ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ based of FFT (quicker)\n elif dirichlet: u ↦ Op(p)(x,D) ⋅ u ≈ u ≈ ∫ eᶦˣᶿ p(x, ξ) 𝓕(u)(ξ) dξ (slower)\n Parameters ---------- u : np.ndarray Input field to which the operator is applied. Should be 1D or 2D depending on the problem dimension. Returns: np.ndarray: Result of applying the pseudo-differential operator to u. Raises: ValueError: If an unsupported boundary condition is specified. """ if not self.is_spatial: return self.apply_psiOp_constant(u) elif self.boundary_condition == 'periodic': return self.apply_psiOp_kohn_nirenberg_fft(u) elif self.boundary_condition == 'dirichlet': return self.apply_psiOp_kohn_nirenberg_nonperiodic(u) else: raise ValueError(f"Invalid boundary condition '{self.boundary_condition}'") def apply_psiOp_constant(self, u)-
Apply a constant-coefficient pseudo-differential operator in Fourier space.
This method assumes the symbol is diagonal in the Fourier basis and acts as a multiplication operator. It performs the operation:
(ψu)(x) = 𝓕⁻¹[ -σ(k) · 𝓕[u](k) ]where: - σ(k) is the combined pseudo-differential operator symbol - 𝓕 denotes the forward Fourier transform - 𝓕⁻¹ denotes the inverse Fourier transform
The dealiasing mask is applied before returning to physical space.
Parameters
u:np.ndarray- Input function in physical space (real-valued or complex-valued)
Returns
np.ndarray- Result of applying the pseudo-differential operator to u, same shape as input
Expand source code
def apply_psiOp_constant(self, u): """ Apply a constant-coefficient pseudo-differential operator in Fourier space. This method assumes the symbol is diagonal in the Fourier basis and acts as a multiplication operator. It performs the operation: (ψu)(x) = 𝓕⁻¹[ -σ(k) · 𝓕[u](k) ] where: - σ(k) is the combined pseudo-differential operator symbol - 𝓕 denotes the forward Fourier transform - 𝓕⁻¹ denotes the inverse Fourier transform The dealiasing mask is applied before returning to physical space. Parameters ---------- u : np.ndarray Input function in physical space (real-valued or complex-valued) Returns: np.ndarray : Result of applying the pseudo-differential operator to u, same shape as input """ u_hat = self.fft(u) u_hat *= -self.combined_symbol u_hat *= self.dealiasing_mask return self.ifft(u_hat) def apply_psiOp_kohn_nirenberg_fft(self, u)-
Apply a pseudo-differential operator using the Kohn–Nirenberg quantization in Fourier space.
This method evaluates the action of a pseudo-differential operator defined by the total symbol, computed from all psiOp terms in the equation. It uses the fast Fourier transform (FFT) for efficiency in periodic domains.
Parameters
u:np.ndarray- Input function in real space to which the operator is applied.
Returns
np.ndarray- Resulting function after applying the pseudo-differential operator.
Process
- Compute the total symbolic expression of the pseudo-differential operator.
- Build a callable numerical function from the symbol.
- Evaluate Op(p)(u) via the Kohn–Nirenberg quantization using FFT.
Note
- Assumes periodic boundary conditions.
- The returned result is the negative of the standard definition due to PDE sign conventions.
Expand source code
def apply_psiOp_kohn_nirenberg_fft(self, u): """ Apply a pseudo-differential operator using the Kohn–Nirenberg quantization in Fourier space. This method evaluates the action of a pseudo-differential operator defined by the total symbol, computed from all psiOp terms in the equation. It uses the fast Fourier transform (FFT) for efficiency in periodic domains. Parameters ---------- u : np.ndarray Input function in real space to which the operator is applied. Returns: np.ndarray: Resulting function after applying the pseudo-differential operator. Process: 1. Compute the total symbolic expression of the pseudo-differential operator. 2. Build a callable numerical function from the symbol. 3. Evaluate Op(p)(u) via the Kohn–Nirenberg quantization using FFT. Note: - Assumes periodic boundary conditions. - The returned result is the negative of the standard definition due to PDE sign conventions. """ total_symbol = self.total_symbol_expr() symbol_func = self.build_symbol_func(total_symbol) return -self.kohn_nirenberg_fft(u_vals=u, symbol_func=symbol_func) def apply_psiOp_kohn_nirenberg_nonperiodic(self, u)-
Apply a pseudo-differential operator using the Kohn–Nirenberg quantization on non-periodic domains.
This method evaluates the action of a pseudo-differential operator Op(p) on a function u via the Kohn–Nirenberg representation. It supports both 1D and 2D cases and uses spatial and frequency grids to evaluate the operator symbol p(x, ξ).
The operator symbol p(x, ξ) is extracted from the PDE and evaluated numerically using
_total_symbol_exprand_build_symbol_func.Parameters
u:np.ndarray- Input function (real space) to which the operator is applied.
Returns
np.ndarray- Result of applying Op(p) to u in real space.
Notes
- For 1D: p(x, ξ) is evaluated over x_grid and xi_grid.
- For 2D: p(x, y, ξ, η) is evaluated over (x_grid, y_grid) and (xi_grid, eta_grid).
- The result is computed using
kohn_nirenberg_nonperiodic, which handles non-periodic boundary conditions.
Expand source code
def apply_psiOp_kohn_nirenberg_nonperiodic(self, u): """ Apply a pseudo-differential operator using the Kohn–Nirenberg quantization on non-periodic domains. This method evaluates the action of a pseudo-differential operator Op(p) on a function u via the Kohn–Nirenberg representation. It supports both 1D and 2D cases and uses spatial and frequency grids to evaluate the operator symbol p(x, ξ). The operator symbol p(x, ξ) is extracted from the PDE and evaluated numerically using `_total_symbol_expr` and `_build_symbol_func`. Parameters ---------- u : np.ndarray Input function (real space) to which the operator is applied. Returns: np.ndarray: Result of applying Op(p) to u in real space. Notes: - For 1D: p(x, ξ) is evaluated over x_grid and xi_grid. - For 2D: p(x, y, ξ, η) is evaluated over (x_grid, y_grid) and (xi_grid, eta_grid). - The result is computed using `kohn_nirenberg_nonperiodic`, which handles non-periodic boundary conditions. """ total_symbol = self.total_symbol_expr() symbol_func = self.build_symbol_func(total_symbol) if self.dim == 1: return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=self.x_grid, xi_grid=self.kx, symbol_func=symbol_func) else: return -self.kohn_nirenberg_nonperiodic(u_vals=u, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=symbol_func) def build_symbol_func(self, expr)-
Build a numerical evaluation function from a symbolic pseudo-differential operator expression.
This method converts a symbolic expression representing a pseudo-differential operator into a callable NumPy-compatible function. The function accepts spatial and frequency variables depending on the dimensionality of the problem.
Parameters
expr:sympy expression- A SymPy expression representing the symbol of the pseudo-differential operator. It may depend on spatial variables (x, y) and frequency variables (xi, eta).
Returns
function-
A lambdified function that takes:
- In 1D:
(x, xi)— spatial coordinate and frequency. - In 2D:
(x, y, xi, eta)— spatial coordinates and frequencies.
Returns a NumPy array of evaluated symbol values over input grids.
- In 1D:
Notes
- Uses
lambdifyfrom SymPy with the'numpy'backend for efficient vectorized evaluation. - Real variable assumptions are enforced to ensure proper behavior in numerical contexts.
- Used internally by methods like
apply_psiOp,evaluate, and visualization tools.
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def build_symbol_func(self, expr): """ Build a numerical evaluation function from a symbolic pseudo-differential operator expression. This method converts a symbolic expression representing a pseudo-differential operator into a callable NumPy-compatible function. The function accepts spatial and frequency variables depending on the dimensionality of the problem. Parameters ---------- expr : sympy expression A SymPy expression representing the symbol of the pseudo-differential operator. It may depend on spatial variables (x, y) and frequency variables (xi, eta). Returns: function : A lambdified function that takes: - In 1D: `(x, xi)` — spatial coordinate and frequency. - In 2D: `(x, y, xi, eta)` — spatial coordinates and frequencies. Returns a NumPy array of evaluated symbol values over input grids. Notes: - Uses `lambdify` from SymPy with the `'numpy'` backend for efficient vectorized evaluation. - Real variable assumptions are enforced to ensure proper behavior in numerical contexts. - Used internally by methods like `apply_psiOp`, `evaluate`, and visualization tools. """ if self.dim == 1: x, xi = symbols('x xi', real=True) return lambdify((x, xi), expr, 'numpy') else: x, y, xi, eta = symbols('x y xi eta', real=True) return lambdify((x, y, xi, eta), expr, 'numpy') def check_cfl_condition(self)-
Check the CFL (Courant–Friedrichs–Lewymann) condition based on group velocity for second-order time-dependent PDEs.
This method verifies whether the chosen time step dt satisfies the numerical stability condition derived from the maximum wave propagation speed in the system. It supports both 1D and 2D problems, with or without a symbolic dispersion relation ω(k).
The CFL condition ensures that information does not propagate further than one grid cell per time step. A safety factor of 0.5 is applied by default to ensure robustness.
Notes:
- In 1D, the group velocity v₉(k) = dω/dk is used to compute the maximum wave speed.
- In 2D, the x- and y-directional group velocities are evaluated independently.
- If no dispersion relation is available, the imaginary part of the linear operator L(k) is used as an approximation for wave speed.
Raises:
NotImplementedError: If the spatial dimension is not 1D or 2D.
Prints:
Warning message if the current time step dt exceeds the CFL-stable limit.
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def check_cfl_condition(self): """ Check the CFL (Courant–Friedrichs–Lewymann) condition based on group velocity for second-order time-dependent PDEs. This method verifies whether the chosen time step dt satisfies the numerical stability condition derived from the maximum wave propagation speed in the system. It supports both 1D and 2D problems, with or without a symbolic dispersion relation ω(k). The CFL condition ensures that information does not propagate further than one grid cell per time step. A safety factor of 0.5 is applied by default to ensure robustness. Notes: - In 1D, the group velocity v₉(k) = dω/dk is used to compute the maximum wave speed. - In 2D, the x- and y-directional group velocities are evaluated independently. - If no dispersion relation is available, the imaginary part of the linear operator L(k) is used as an approximation for wave speed. Raises: ------- NotImplementedError: If the spatial dimension is not 1D or 2D. Prints: ------- Warning message if the current time step dt exceeds the CFL-stable limit. """ print("\n*****************") print("* CFL condition *") print("*****************\n") cfl_factor = 0.5 # Safety factor if self.dim == 1: if self.temporal_order == 2 and hasattr(self, 'omega'): k_vals = self.kx omega_vals = np.real(self.omega(k_vals)) with np.errstate(divide='ignore', invalid='ignore'): v_group = np.gradient(omega_vals, k_vals) max_speed = np.max(np.abs(v_group)) else: max_speed = np.max(np.abs(np.imag(self.L(self.kx)))) dx = self.Lx / self.Nx cfl_limit = cfl_factor * dx / max_speed if max_speed != 0 else np.inf if self.dt > cfl_limit: print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}") elif self.dim == 2: if self.temporal_order == 2 and hasattr(self, 'omega'): k_vals = self.kx omega_x = np.real(self.omega(k_vals, 0)) omega_y = np.real(self.omega(0, k_vals)) with np.errstate(divide='ignore', invalid='ignore'): v_group_x = np.gradient(omega_x, k_vals) v_group_y = np.gradient(omega_y, k_vals) max_speed_x = np.max(np.abs(v_group_x)) max_speed_y = np.max(np.abs(v_group_y)) else: max_speed_x = np.max(np.abs(np.imag(self.L(self.kx, 0)))) max_speed_y = np.max(np.abs(np.imag(self.L(0, self.ky)))) dx = self.Lx / self.Nx dy = self.Ly / self.Ny cfl_limit = cfl_factor / (max_speed_x / dx + max_speed_y / dy) if (max_speed_x + max_speed_y) != 0 else np.inf if self.dt > cfl_limit: print(f"CFL condition violated: dt = {self.dt}, max allowed dt = {cfl_limit}") else: raise NotImplementedError("Only 1D and 2D problems are supported.") def check_symbol_conditions(self, k_range=None, verbose=True)-
Check strict analytic conditions on the linear symbol self.L_symbolic: This method evaluates three key properties of the Fourier multiplier symbol a(k) = self.L(k), which are crucial for well-posedness, stability, and numerical efficiency. The checks apply to both 1D and 2D cases.
Conditions checked:
-
Stability condition: Re(a(k)) ≤ 0 for all k ≠ 0 Ensures that the system does not exhibit exponential growth in time.
-
Dissipation condition: Re(a(k)) ≤ -δ |k|² for large |k| Ensures sufficient damping at high frequencies to avoid oscillatory instability.
-
Growth condition: |a(k)| ≤ C (1 + |k|)^m with m ≤ 4 Ensures that the symbol does not grow too rapidly with frequency, which would otherwise cause numerical instability or unphysical amplification.
Parameters
k_range:tupleorNone, optional- Specifies the range of frequencies to test in the form (k_min, k_max, N). If None, defaults are used: [-10, 10] with 500 points in 1D, or [-10, 10] with 100 points per axis in 2D.
verbose:bool, default=True- If True, prints detailed results of each condition check.
Returns:
None Output is printed directly to the console for interpretability.
Notes:
- In 2D, the radial frequency |k| = √(kx² + ky²) is used for comparisons.
- The dissipation threshold assumes δ = 0.01 and p = 2 by default.
- The growth ratio is compared against |k|⁴; values above 100 indicate rapid growth.
- This function is typically called during solver setup or analysis phase.
See Also:
analyze_wave_propagation : For further symbolic and numerical analysis of dispersion. plot_symbol : Visualizes the symbol's behavior over the frequency domain.
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def check_symbol_conditions(self, k_range=None, verbose=True): """ Check strict analytic conditions on the linear symbol self.L_symbolic: This method evaluates three key properties of the Fourier multiplier symbol a(k) = self.L(k), which are crucial for well-posedness, stability, and numerical efficiency. The checks apply to both 1D and 2D cases. Conditions checked: ------------------ 1. **Stability condition**: Re(a(k)) ≤ 0 for all k ≠ 0 Ensures that the system does not exhibit exponential growth in time. 2. **Dissipation condition**: Re(a(k)) ≤ -δ |k|² for large |k| Ensures sufficient damping at high frequencies to avoid oscillatory instability. 3. **Growth condition**: |a(k)| ≤ C (1 + |k|)^m with m ≤ 4 Ensures that the symbol does not grow too rapidly with frequency, which would otherwise cause numerical instability or unphysical amplification. Parameters ---------- k_range : tuple or None, optional Specifies the range of frequencies to test in the form (k_min, k_max, N). If None, defaults are used: [-10, 10] with 500 points in 1D, or [-10, 10] with 100 points per axis in 2D. verbose : bool, default=True If True, prints detailed results of each condition check. Returns: -------- None Output is printed directly to the console for interpretability. Notes: ------ - In 2D, the radial frequency |k| = √(kx² + ky²) is used for comparisons. - The dissipation threshold assumes δ = 0.01 and p = 2 by default. - The growth ratio is compared against |k|⁴; values above 100 indicate rapid growth. - This function is typically called during solver setup or analysis phase. See Also: --------- analyze_wave_propagation : For further symbolic and numerical analysis of dispersion. plot_symbol : Visualizes the symbol's behavior over the frequency domain. """ print("\n********************") print("* Symbol condition *") print("********************\n") if self.dim == 1: if k_range is None: k_vals = np.linspace(-10, 10, 500) else: k_min, k_max, N = k_range k_vals = np.linspace(k_min, k_max, N) L_vals = self.L(k_vals) k_abs = np.abs(k_vals) elif self.dim == 2: if k_range is None: k_vals = np.linspace(-10, 10, 100) else: k_min, k_max, N = k_range k_vals = np.linspace(k_min, k_max, N) KX, KY = np.meshgrid(k_vals, k_vals) L_vals = self.L(KX, KY) k_abs = np.sqrt(KX**2 + KY**2) else: raise ValueError("Only 1D and 2D dimensions are supported.") re_vals = np.real(L_vals) abs_vals = np.abs(L_vals) # === Condition 1: Stability if np.any(re_vals > 1e-12): max_pos = np.max(re_vals) if verbose: print(f"❌ Stability violated: max Re(a(k)) = {max_pos}") print("Unstable symbol: Re(a(k)) > 0") elif verbose: print("✅ Spectral stability satisfied: Re(a(k)) ≤ 0") # === Condition 2: Dissipation mask = k_abs > 2 if np.any(mask): re_decay = re_vals[mask] expected_decay = -0.01 * k_abs[mask]**2 if np.any(re_decay > expected_decay + 1e-6): if verbose: print("⚠️ Insufficient high-frequency dissipation") else: if verbose: print("✅ Proper high-frequency dissipation") # === Condition 3: Growth growth_ratio = abs_vals / (1 + k_abs)**4 if np.max(growth_ratio) > 100: if verbose: print("⚠️ Symbol grows rapidly: |a(k)| ≳ |k|^4") else: if verbose: print("✅ Reasonable spectral growth") if verbose: print("✔ Symbol analysis completed.") -
def compute_energy(self)-
Compute the total energy of the wave equation solution for second-order temporal PDEs. The energy is defined as: E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹ᐟ²u|² ] dx where L is the linear operator associated with the spatial part of the PDE, and L¹ᐟ² denotes its square root in Fourier space.
This method supports both 1D and 2D problems and is only meaningful when self.temporal_order == 2 (second-order time derivative).
Returns
floatorNone:- Total energy at current time step. Returns None if the temporal order is not 2 or if no valid velocity data (v_prev) is available.
Notes
- Uses FFT-based spectral differentiation to compute the spatial contributions.
- Assumes periodic boundary conditions.
- Handles both real and complex-valued solutions.
Expand source code
def compute_energy(self): """ Compute the total energy of the wave equation solution for second-order temporal PDEs. The energy is defined as: E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹ᐟ²u|² ] dx where L is the linear operator associated with the spatial part of the PDE, and L¹ᐟ² denotes its square root in Fourier space. This method supports both 1D and 2D problems and is only meaningful when self.temporal_order == 2 (second-order time derivative). Returns ------- float or None: Total energy at current time step. Returns None if the temporal order is not 2 or if no valid velocity data (v_prev) is available. Notes ----- - Uses FFT-based spectral differentiation to compute the spatial contributions. - Assumes periodic boundary conditions. - Handles both real and complex-valued solutions. """ if self.temporal_order != 2 or self.v_prev is None: return None u = self.u_prev v = self.v_prev # Fourier transform of u u_hat = self.fft(u) if self.dim == 1: # 1D case L_vals = self.L(self.KX) sqrt_L = np.sqrt(np.abs(L_vals)) Lu_hat = sqrt_L * u_hat # Apply sqrt(|L(k)|) in Fourier space Lu = self.ifft(Lu_hat) dx = self.Lx / self.Nx energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2) total_energy = np.sum(energy_density) * dx elif self.dim == 2: # 2D case L_vals = self.L(self.KX, self.KY) sqrt_L = np.sqrt(np.abs(L_vals)) Lu_hat = sqrt_L * u_hat Lu = self.ifft(Lu_hat) dx = self.Lx / self.Nx dy = self.Ly / self.Ny energy_density = 0.5 * (np.abs(v)**2 + np.abs(Lu)**2) total_energy = np.sum(energy_density) * dx * dy else: raise ValueError("Unsupported dimension for u.") return total_energy def compute_linear_operator(self)-
Compute the symbolic Fourier representation L(k) of the linear operator derived from the linear part of the PDE.
This method constructs a dispersion relation by applying each symbolic derivative to a plane wave exp(i(k·x - ωt)) and extracting the resulting expression. It handles arbitrary derivative combinations and includes symbolic and pseudo-differential terms.
Steps:
- Construct a plane wave φ(x, t) = exp(i(k·x - ωt)).
- Apply each term from self.linear_terms to φ.
- Normalize by φ and simplify to obtain L(k).
- Include symbolic terms (e.g., psiOp) if present.
- Detect the temporal order from the dispersion relation.
- Build the numerical function L(k) via lambdify.
Sets:
- self.L_symbolic : sympy.Expr Symbolic form of L(k).
- self.L : callable Numerical function of L(kx[, ky]).
- self.omega : callable or None Frequency root ω(k), if available.
- self.temporal_order : int Order of time derivatives detected.
- self.psi_ops : list of (coeff, PseudoDifferentialOperator) Pseudo-differential terms present in the equation.
Raises:
ValueError if the dimension is unsupported or the dispersion relation fails.
Expand source code
def compute_linear_operator(self): """ Compute the symbolic Fourier representation L(k) of the linear operator derived from the linear part of the PDE. This method constructs a dispersion relation by applying each symbolic derivative to a plane wave exp(i(k·x - ωt)) and extracting the resulting expression. It handles arbitrary derivative combinations and includes symbolic and pseudo-differential terms. Steps: ------- 1. Construct a plane wave φ(x, t) = exp(i(k·x - ωt)). 2. Apply each term from self.linear_terms to φ. 3. Normalize by φ and simplify to obtain L(k). 4. Include symbolic terms (e.g., psiOp) if present. 5. Detect the temporal order from the dispersion relation. 6. Build the numerical function L(k) via lambdify. Sets: ----- - self.L_symbolic : sympy.Expr Symbolic form of L(k). - self.L : callable Numerical function of L(kx[, ky]). - self.omega : callable or None Frequency root ω(k), if available. - self.temporal_order : int Order of time derivatives detected. - self.psi_ops : list of (coeff, PseudoDifferentialOperator) Pseudo-differential terms present in the equation. Raises: ------- ValueError if the dimension is unsupported or the dispersion relation fails. """ print("\n*******************************") print("* Linear operator computation *") print("*******************************\n") # --- Step 1: symbolic variables --- omega = symbols("omega") if self.dim == 1: kvars = [symbols("kx")] space_vars = [self.x] elif self.dim == 2: kvars = symbols("kx ky") space_vars = [self.x, self.y] else: raise ValueError("Only 1D and 2D are supported.") kdict = dict(zip(space_vars, kvars)) self.k_symbols = kvars # Plane wave expression phase = sum(k * x for k, x in zip(kvars, space_vars)) - omega * self.t plane_wave = exp(I * phase) # --- Step 2: build lhs expression from linear terms --- lhs = 0 for deriv, coeff in self.linear_terms.items(): if isinstance(deriv, Derivative): total_factor = 1 for var, n in deriv.variable_count: if var == self.t: total_factor *= (-I * omega)**n elif var in kdict: total_factor *= (I * kdict[var])**n else: raise ValueError(f"Unknown variable {var} in derivative") lhs += coeff * total_factor * plane_wave elif deriv == self.u: lhs += coeff * plane_wave else: raise ValueError(f"Unsupported linear term: {deriv}") # --- Step 3: dispersion relation --- equation = simplify(lhs / plane_wave) print("\nCharacteristic equation before symbol treatment:") pprint(equation) print("\n--- Symbolic symbol analysis ---") symb_omega = 0 symb_k = 0 for coeff, symbol in self.symbol_terms: if symbol.has(omega): # Ajouter directement les termes dépendant de omega symb_omega += coeff * symbol elif any(symbol.has(k) for k in self.k_symbols): symb_k += coeff * symbol.subs(dict(zip(symbol.free_symbols, self.k_symbols))) print(f"symb_omega: {symb_omega}") print(f"symb_k: {symb_k}") equation = equation + symb_omega + symb_k print("\nRaw characteristic equation:") pprint(equation) # Temporal derivative order detection try: poly_eq = Eq(equation, 0) poly = poly_eq.lhs.as_poly(omega) self.temporal_order = poly.degree() if poly else 0 except Exception as e: warnings.warn(f"Could not determine temporal order: {e}", RuntimeWarning) self.temporal_order = 0 print(f"Temporal order from dispersion relation: {self.temporal_order}") print('self.pseudo_terms = ', self.pseudo_terms) if self.pseudo_terms: coeff_time = 1 for term, coeff in self.linear_terms.items(): if isinstance(term, Derivative) and any(var == self.t for var, _ in term.variable_count): coeff_time = coeff print(f"✅ Time derivative coefficient detected: {coeff_time}") self.psi_ops = [] for coeff, sym_expr in self.pseudo_terms: # expr est le Sympy expr. différentiel, var_x la liste [x] ou [x,y] psi = PseudoDifferentialOperator(sym_expr / coeff_time, self.spatial_vars, self.u, mode='symbol') self.psi_ops.append((coeff, psi)) else: dispersion = solve(Eq(equation, 0), omega) if not dispersion: raise ValueError("No solution found for omega") print("\n--- Solutions found ---") pprint(dispersion) if self.temporal_order == 2: omega_expr = simplify(sqrt(dispersion[0]**2)) self.omega_symbolic = omega_expr self.omega = lambdify(self.k_symbols, omega_expr, "numpy") self.L_symbolic = -omega_expr**2 else: self.L_symbolic = -I * dispersion[0] self.L = lambdify(self.k_symbols, self.L_symbolic, "numpy") print("\n--- Final linear operator ---") pprint(self.L_symbolic) def evaluate_source_at_t0(self)-
Evaluate source terms at initial time t = 0 over the spatial grid.
This private method computes the total contribution of all source terms at the initial time, evaluated across the entire spatial domain. It supports both one-dimensional (1D) and two-dimensional (2D) configurations.
Returns
np.ndarray- A numpy array representing the evaluated source term at t=0:
- In 1D: Shape (Nx,), evaluated at each x in
self.x_grid. - In 2D: Shape (Nx, Ny), evaluated at each (x, y) pair in the grid.
Notes
- The symbolic expressions in
self.source_termsare substituted with numerical values at t=0. - In 1D, each term is evaluated at (t=0, x=x_val).
- In 2D, each term is evaluated at (t=0, x=x_val, y=y_val).
- Evaluated using SymPy's
evalf()to ensure numeric conversion. - This method assumes that the source terms have already been lambdified or are compatible with symbolic substitution.
See Also
setup- Initializes the spatial grid and source terms.
solve- Uses this evaluation during the first time step.
Expand source code
def evaluate_source_at_t0(self): """ Evaluate source terms at initial time t = 0 over the spatial grid. This private method computes the total contribution of all source terms at the initial time, evaluated across the entire spatial domain. It supports both one-dimensional (1D) and two-dimensional (2D) configurations. Returns ------- np.ndarray A numpy array representing the evaluated source term at t=0: - In 1D: Shape (Nx,), evaluated at each x in `self.x_grid`. - In 2D: Shape (Nx, Ny), evaluated at each (x, y) pair in the grid. Notes ----- - The symbolic expressions in `self.source_terms` are substituted with numerical values at t=0. - In 1D, each term is evaluated at (t=0, x=x_val). - In 2D, each term is evaluated at (t=0, x=x_val, y=y_val). - Evaluated using SymPy's `evalf()` to ensure numeric conversion. - This method assumes that the source terms have already been lambdified or are compatible with symbolic substitution. See Also -------- setup : Initializes the spatial grid and source terms. solve : Uses this evaluation during the first time step. """ if self.dim == 1: # Evaluation on the 1D spatial grid return np.array([ sum(term.subs(self.t, 0).subs(self.x, x_val).evalf() for term in self.source_terms) for x_val in self.x_grid ], dtype=np.float64) else: # Evaluation on the 2D spatial grid return np.array([ [sum(term.subs({self.t: 0, self.x: x_val, self.y: y_val}).evalf() for term in self.source_terms) for y_val in self.y_grid] for x_val in self.x_grid ], dtype=np.float64) def initialize_conditions(self, initial_condition, initial_velocity)-
Initialize the solution and velocity fields at t = 0.
This private method sets up the initial state of the solution
u_prevand, if applicable, the time derivative (velocity)v_prevfor second-order evolution equations.For second-order equations, it also computes the backward-in-time value
u_prev2needed by the Leap-Frog method. The acceleration at t = 0 is computed from: ∂ₜ²u = L(u) + N(u) + f(x, t=0) where L is the linear operator, N is the nonlinear term, and f is the source term.Parameters
initial_condition:callable- Function returning the initial condition u(x, 0) or u(x, y, 0).
initial_velocity:callableorNone- Function returning the initial velocity ∂ₜu(x, 0) or ∂ₜu(x, y, 0). Required for second-order equations; ignored otherwise.
Raises
ValueError- If
initial_velocityis not provided for second-order equations.
Notes
- Applies periodic boundary conditions after setting initial data.
- Stores a copy of the initial state in
self.framesfor visualization/output. - In second-order systems, initializes
self.u_prev2using a Taylor expansion: u_prev2 = u_prev - dt * v_prev + 0.5 * dt² * (∂ₜ²u)
See Also
apply_boundary- Enforces periodic boundary conditions on the solution field.
psiOp_apply- Computes pseudo-differential operator action for acceleration.
linear_rhs- Evaluates linear part of the equation in Fourier space.
apply_nonlinear- Handles nonlinear terms with spectral differentiation.
evaluate_source_at_t0- Evaluates source terms at the initial time.
Expand source code
def initialize_conditions(self, initial_condition, initial_velocity): """ Initialize the solution and velocity fields at t = 0. This private method sets up the initial state of the solution `u_prev` and, if applicable, the time derivative (velocity) `v_prev` for second-order evolution equations. For second-order equations, it also computes the backward-in-time value `u_prev2` needed by the Leap-Frog method. The acceleration at t = 0 is computed from: ∂ₜ²u = L(u) + N(u) + f(x, t=0) where L is the linear operator, N is the nonlinear term, and f is the source term. Parameters ---------- initial_condition : callable Function returning the initial condition u(x, 0) or u(x, y, 0). initial_velocity : callable or None Function returning the initial velocity ∂ₜu(x, 0) or ∂ₜu(x, y, 0). Required for second-order equations; ignored otherwise. Raises ------ ValueError If `initial_velocity` is not provided for second-order equations. Notes ----- - Applies periodic boundary conditions after setting initial data. - Stores a copy of the initial state in `self.frames` for visualization/output. - In second-order systems, initializes `self.u_prev2` using a Taylor expansion: u_prev2 = u_prev - dt * v_prev + 0.5 * dt² * (∂ₜ²u) See Also -------- apply_boundary : Enforces periodic boundary conditions on the solution field. psiOp_apply : Computes pseudo-differential operator action for acceleration. linear_rhs : Evaluates linear part of the equation in Fourier space. apply_nonlinear : Handles nonlinear terms with spectral differentiation. evaluate_source_at_t0 : Evaluates source terms at the initial time. """ # Initial condition if self.dim == 1: self.u_prev = initial_condition(self.X) else: self.u_prev = initial_condition(self.X, self.Y) self.apply_boundary(self.u_prev) # Initial velocity (second order) if self.temporal_order == 2: if initial_velocity is None: raise ValueError("Initial velocity is required for second-order equations.") if self.dim == 1: self.v_prev = initial_velocity(self.X) else: self.v_prev = initial_velocity(self.X, self.Y) self.u0 = np.copy(self.u_prev) self.v0 = np.copy(self.v_prev) # Calculation of u_prev2 (initial acceleration) if not hasattr(self, 'u_prev2'): if self.has_psi: acc0 = self.apply_psiOp(self.u_prev) else: acc0 = self.linear_rhs(self.u_prev, is_v=False) rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False) acc0 += rhs_nl if hasattr(self, 'source_terms') and self.source_terms: acc0 += self.evaluate_source_at_t0() self.u_prev2 = self.u_prev - self.dt * self.v_prev + 0.5 * self.dt**2 * acc0 self.frames = [self.u_prev.copy()] def kohn_nirenberg_fft(self, u_vals, symbol_func, freq_window='gaussian', clamp=1000000.0, space_window=False)-
Numerically stable Kohn–Nirenberg quantization of a pseudo-differential operator.
Applies the pseudo-differential operator Op(p) to the function f via the Kohn–Nirenberg quantization:
[Op(p)f](x) = (1/(2π)^d) ∫ p(x, ξ) e^{ix·ξ} ℱ[f](ξ) dξwhere p(x, ξ) is a symbol that may depend on both spatial variables x and frequency variables ξ.
This method supports both 1D and 2D cases and includes optional smoothing techniques to improve numerical stability.
Parameters
u_vals:np.ndarray- Spatial samples of the input function f(x) or f(x, y), defined on a uniform grid.
symbol_func:callable- A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array.
freq_window:{'gaussian', 'hann', None}, optional- Type of frequency-domain window to apply: - 'gaussian': smooth decay near high frequencies - 'hann': cosine-based tapering with hard cutoff - None: no frequency window applied
clamp:float, optional- Upper bound on the absolute value of the symbol. Prevents numerical blow-up from large values.
space_window:bool, optional- Whether to apply a spatial Gaussian window to suppress edge effects in physical space.
Returns
np.ndarray- The result of applying the pseudo-differential operator to f, returned as a real or complex array of the same shape as u_vals.
Notes
- The implementation uses FFT-based quadrature of the inverse Fourier transform.
- Symbol evaluation is vectorized over spatial and frequency grids.
- Frequency and spatial windows help mitigate oscillatory behavior and aliasing.
- In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η).
Expand source code
def kohn_nirenberg_fft(self, u_vals, symbol_func, freq_window='gaussian', clamp=1e6, space_window=False): """ Numerically stable Kohn–Nirenberg quantization of a pseudo-differential operator. Applies the pseudo-differential operator Op(p) to the function f via the Kohn–Nirenberg quantization: [Op(p)f](x) = (1/(2π)^d) ∫ p(x, ξ) e^{ix·ξ} ℱ[f](ξ) dξ where p(x, ξ) is a symbol that may depend on both spatial variables x and frequency variables ξ. This method supports both 1D and 2D cases and includes optional smoothing techniques to improve numerical stability. Parameters ---------- u_vals : np.ndarray Spatial samples of the input function f(x) or f(x, y), defined on a uniform grid. symbol_func : callable A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array. freq_window : {'gaussian', 'hann', None}, optional Type of frequency-domain window to apply: - 'gaussian': smooth decay near high frequencies - 'hann': cosine-based tapering with hard cutoff - None: no frequency window applied clamp : float, optional Upper bound on the absolute value of the symbol. Prevents numerical blow-up from large values. space_window : bool, optional Whether to apply a spatial Gaussian window to suppress edge effects in physical space. Returns ------- np.ndarray The result of applying the pseudo-differential operator to f, returned as a real or complex array of the same shape as u_vals. Notes ----- - The implementation uses FFT-based quadrature of the inverse Fourier transform. - Symbol evaluation is vectorized over spatial and frequency grids. - Frequency and spatial windows help mitigate oscillatory behavior and aliasing. - In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η). """ # === Common setup === xg = self.x_grid dx = xg[1] - xg[0] if self.dim == 1: # === 1D case === # Frequency grid (shifted to center zero) Nx = self.Nx k = 2 * np.pi * fftshift(fftfreq(Nx, d=dx)) dk = k[1] - k[0] # Centered FFT of input f_shift = fftshift(u_vals) f_hat = self.fft(f_shift) * dx f_hat = fftshift(f_hat) # Build meshgrid for (x, ξ) X, K = np.meshgrid(xg, k, indexing='ij') # Evaluate the symbol p(x, ξ) P = symbol_func(X, K) # Optional: clamp extreme values P = np.clip(P, -clamp, clamp) # === Frequency-domain window === if freq_window == 'gaussian': sigma = 0.8 * np.max(np.abs(k)) W = np.exp(-(K / sigma) ** 4) P *= W elif freq_window == 'hann': W = 0.5 * (1 + np.cos(np.pi * K / np.max(np.abs(K)))) P *= W * (np.abs(K) < np.max(np.abs(K))) # === Optional spatial window === if space_window: x0 = (xg[0] + xg[-1]) / 2 L = (xg[-1] - xg[0]) / 2 S = np.exp(-((X - x0) / L) ** 2) P *= S # === Oscillatory kernel and integration === kernel = np.exp(1j * X * K) integrand = P * f_hat[None, :] * kernel # Approximate inverse Fourier integral u = np.sum(integrand, axis=1) * dk / (2 * np.pi) return u else: # === 2D case === yg = self.y_grid dy = yg[1] - yg[0] Nx, Ny = self.Nx, self.Ny # Frequency grids kx = 2 * np.pi * fftshift(fftfreq(Nx, d=dx)) ky = 2 * np.pi * fftshift(fftfreq(Ny, d=dy)) dkx = kx[1] - kx[0] dky = ky[1] - ky[0] # 2D FFT of f(x, y) f_shift = fftshift(u_vals) f_hat = self.fft(f_shift) * dx * dy f_hat = fftshift(f_hat) # Create 4D grids for broadcasting X, Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij') KX, KY = np.meshgrid(kx, ky, indexing='ij') Xb = X[:, :, None, None] Yb = Y[:, :, None, None] KXb = KX[None, None, :, :] KYb = KY[None, None, :, :] # Evaluate p(x, y, ξ, η) P_vals = symbol_func(Xb, Yb, KXb, KYb) P_vals = np.clip(P_vals, -clamp, clamp) # === Frequency windowing === if freq_window == 'gaussian': sigma_kx = 0.8 * np.max(np.abs(kx)) sigma_ky = 0.8 * np.max(np.abs(ky)) W_kx = np.exp(-(KXb / sigma_kx) ** 4) W_ky = np.exp(-(KYb / sigma_ky) ** 4) P_vals *= W_kx * W_ky elif freq_window == 'hann': Wx = 0.5 * (1 + np.cos(np.pi * KXb / np.max(np.abs(kx)))) Wy = 0.5 * (1 + np.cos(np.pi * KYb / np.max(np.abs(ky)))) mask_x = np.abs(KXb) < np.max(np.abs(kx)) mask_y = np.abs(KYb) < np.max(np.abs(ky)) P_vals *= Wx * Wy * mask_x * mask_y # === Optional spatial tapering === if space_window: x0 = (self.x_grid[0] + self.x_grid[-1]) / 2 y0 = (self.y_grid[0] + self.y_grid[-1]) / 2 Lx = (self.x_grid[-1] - self.x_grid[0]) / 2 Ly = (self.y_grid[-1] - self.y_grid[0]) / 2 S = np.exp(-((Xb - x0) / Lx) ** 2 - ((Yb - y0) / Ly) ** 2) P_vals *= S # === Oscillatory kernel and integration === phase = np.exp(1j * (Xb * KXb + Yb * KYb)) integrand = P_vals * phase * f_hat[None, None, :, :] # 2D Fourier inversion (numerical integration) u = np.sum(integrand, axis=(2, 3)) * dkx * dky / (2 * np.pi) ** 2 return u def kohn_nirenberg_nonperiodic(self, u_vals, x_grid, xi_grid, symbol_func, freq_window='gaussian', clamp=1000000.0, space_window=False)-
Numerically applies the Kohn–Nirenberg quantization of a pseudo-differential operator in a non-periodic setting.
This method computes:
Op(p)u = (1/(2π)^d) ∫ p(x, ξ) e^{i x·ξ} ℱu dξ
where p(x, ξ) is a general symbol that may depend on both spatial and frequency variables. It supports both 1D and 2D inputs and includes optional numerical smoothing techniques to enhance stability for non-smooth or oscillatory symbols.
Parameters
u_vals:np.ndarray- Input function values defined on a uniform spatial grid. Can be 1D (Nx,) or 2D (Nx, Ny).
x_grid:np.ndarray- Spatial grid points along each axis. In 1D: shape (Nx,). In 2D: tuple of two arrays (X, Y) or list of coordinate arrays.
xi_grid:np.ndarray- Frequency grid points. In 1D: shape (Nxi,). In 2D: tuple of two arrays (Xi, Eta) or list of frequency arrays.
symbol_func:callable- A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array.
freq_window:{'gaussian', 'hann', None}, optional-
Type of frequency-domain window to apply for regularization:
- 'gaussian': Smooth exponential decay near high frequencies.
- 'hann': Cosine-based tapering with hard cutoff.
- None: No frequency window applied.
clamp:float, optional- Maximum absolute value allowed for the symbol to prevent numerical overflow. Default is 1e6.
space_window:bool, optional- If True, applies a smooth spatial Gaussian window centered in the domain to reduce boundary artifacts. Default is False.
Returns
np.ndarray- The result of applying the pseudo-differential operator Op(p) to u. Shape matches u_vals.
Notes
- This version does not assume periodicity and is suitable for Dirichlet or Neumann boundary conditions.
- In 1D, the integral is evaluated as a sum over (x, ξ), using matrix exponentials.
- In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η), which can be computationally intensive.
- Symbol evaluation should be vectorized for performance.
- For large grids, consider reducing resolution via resampling before calling this function.
See Also
kohn_nirenberg_fft- Faster implementation for periodic domains using FFT.
PseudoDifferentialOperator- Class for symbolic manipulation of pseudo-differential operators.
Expand source code
def kohn_nirenberg_nonperiodic(self, u_vals, x_grid, xi_grid, symbol_func, freq_window='gaussian', clamp=1e6, space_window=False): """ Numerically applies the Kohn–Nirenberg quantization of a pseudo-differential operator in a non-periodic setting. This method computes: [Op(p)u](x) = (1/(2π)^d) ∫ p(x, ξ) e^{i x·ξ} ℱ[u](ξ) dξ where p(x, ξ) is a general symbol that may depend on both spatial and frequency variables. It supports both 1D and 2D inputs and includes optional numerical smoothing techniques to enhance stability for non-smooth or oscillatory symbols. Parameters ---------- u_vals : np.ndarray Input function values defined on a uniform spatial grid. Can be 1D (Nx,) or 2D (Nx, Ny). x_grid : np.ndarray Spatial grid points along each axis. In 1D: shape (Nx,). In 2D: tuple of two arrays (X, Y) or list of coordinate arrays. xi_grid : np.ndarray Frequency grid points. In 1D: shape (Nxi,). In 2D: tuple of two arrays (Xi, Eta) or list of frequency arrays. symbol_func : callable A function representing the full symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. Must accept NumPy-compatible array inputs and return a complex-valued array. freq_window : {'gaussian', 'hann', None}, optional Type of frequency-domain window to apply for regularization: - 'gaussian': Smooth exponential decay near high frequencies. - 'hann': Cosine-based tapering with hard cutoff. - None: No frequency window applied. clamp : float, optional Maximum absolute value allowed for the symbol to prevent numerical overflow. Default is 1e6. space_window : bool, optional If True, applies a smooth spatial Gaussian window centered in the domain to reduce boundary artifacts. Default is False. Returns ------- np.ndarray The result of applying the pseudo-differential operator Op(p) to u. Shape matches u_vals. Notes ----- - This version does not assume periodicity and is suitable for Dirichlet or Neumann boundary conditions. - In 1D, the integral is evaluated as a sum over (x, ξ), using matrix exponentials. - In 2D, the integration is performed over a 4D tensor product grid (x, y, ξ, η), which can be computationally intensive. - Symbol evaluation should be vectorized for performance. - For large grids, consider reducing resolution via resampling before calling this function. See Also -------- kohn_nirenberg_fft : Faster implementation for periodic domains using FFT. PseudoDifferentialOperator : Class for symbolic manipulation of pseudo-differential operators. """ if u_vals.ndim == 1: # === 1D case === x = x_grid xi = xi_grid dx = x[1] - x[0] dxi = xi[1] - xi[0] phase_ft = np.exp(-1j * np.outer(xi, x)) # (Nxi, Nx) u_hat = dx * np.dot(phase_ft, u_vals) # (Nxi,) X, XI = np.meshgrid(x, xi, indexing='ij') # (Nx, Nxi) sigma_vals = symbol_func(X, XI) # Clamp values sigma_vals = np.clip(sigma_vals, -clamp, clamp) # Frequency window if freq_window == 'gaussian': sigma = 0.8 * np.max(np.abs(XI)) window = np.exp(-(XI / sigma)**4) sigma_vals *= window elif freq_window == 'hann': window = 0.5 * (1 + np.cos(np.pi * XI / np.max(np.abs(XI)))) sigma_vals *= window * (np.abs(XI) < np.max(np.abs(XI))) # Spatial window if space_window: x_center = (x[0] + x[-1]) / 2 L = (x[-1] - x[0]) / 2 window = np.exp(-((X - x_center)/L)**2) sigma_vals *= window exp_matrix = np.exp(1j * np.outer(x, xi)) # (Nx, Nxi) integrand = sigma_vals * u_hat[np.newaxis, :] * exp_matrix result = dxi * np.sum(integrand, axis=1) / (2 * np.pi) return result elif u_vals.ndim == 2: # === 2D case === x1, x2 = x_grid xi1, xi2 = xi_grid dx1 = x1[1] - x1[0] dx2 = x2[1] - x2[0] dxi1 = xi1[1] - xi1[0] dxi2 = xi2[1] - xi2[0] X1, X2 = np.meshgrid(x1, x2, indexing='ij') XI1, XI2 = np.meshgrid(xi1, xi2, indexing='ij') # Fourier transform of u(x1, x2) phase_ft = np.exp(-1j * (np.tensordot(x1, xi1, axes=0)[:, None, :, None] + np.tensordot(x2, xi2, axes=0)[None, :, None, :])) u_hat = np.tensordot(u_vals, phase_ft, axes=([0,1], [0,1])) * dx1 * dx2 # Symbol evaluation sigma_vals = symbol_func(X1[:, :, None, None], X2[:, :, None, None], XI1[None, None, :, :], XI2[None, None, :, :]) # Clamp values sigma_vals = np.clip(sigma_vals, -clamp, clamp) # Frequency window if freq_window == 'gaussian': sigma_xi1 = 0.8 * np.max(np.abs(XI1)) sigma_xi2 = 0.8 * np.max(np.abs(XI2)) window = np.exp(-(XI1[None, None, :, :] / sigma_xi1)**4 - (XI2[None, None, :, :] / sigma_xi2)**4) sigma_vals *= window elif freq_window == 'hann': # Frequency window - Hanning wx = 0.5 * (1 + np.cos(np.pi * XI1 / np.max(np.abs(XI1)))) wy = 0.5 * (1 + np.cos(np.pi * XI2 / np.max(np.abs(XI2)))) # Mask to zero outside max frequency mask_x = (np.abs(XI1) < np.max(np.abs(XI1))) mask_y = (np.abs(XI2) < np.max(np.abs(XI2))) # Expand wx and wy to match sigma_vals shape: (64, 64, 64, 64) sigma_vals *= wx[:, :, None, None] * wy[:, :, None, None] sigma_vals *= mask_x[:, :, None, None] * mask_y[:, :, None, None] # Spatial window if space_window: x_center = (x1[0] + x1[-1])/2 y_center = (x2[0] + x2[-1])/2 Lx = (x1[-1] - x1[0])/2 Ly = (x2[-1] - x2[0])/2 window = np.exp(-((X1 - x_center)/Lx)**2 - ((X2 - y_center)/Ly)**2) sigma_vals *= window[:, :, None, None] # Oscillatory phase phase = np.exp(1j * (X1[:, :, None, None] * XI1[None, None, :, :] + X2[:, :, None, None] * XI2[None, None, :, :])) integrand = sigma_vals * u_hat[None, None, :, :] * phase result = dxi1 * dxi2 * np.sum(integrand, axis=(2, 3)) / (2 * np.pi)**2 return result else: raise NotImplementedError("Only 1D and 2D supported") def linear_rhs(self, u, is_v=False)-
Apply the linear operator (in Fourier space) to the field u or v.
Parameters
u:np.ndarray- Input solution array.
is_v:bool- Whether to apply the operator to v instead of u.
Returns
np.ndarray- Result of applying the linear operator.
Expand source code
def linear_rhs(self, u, is_v=False): """ Apply the linear operator (in Fourier space) to the field u or v. Parameters ---------- u : np.ndarray Input solution array. is_v : bool Whether to apply the operator to v instead of u. Returns ------- np.ndarray Result of applying the linear operator. """ if self.dim == 1: self.symbol_u = np.array(self.L(self.KX), dtype=np.complex128) self.symbol_v = self.symbol_u # même opérateur pour u et v elif self.dim == 2: self.symbol_u = np.array(self.L(self.KX, self.KY), dtype=np.complex128) self.symbol_v = self.symbol_u u_hat = self.fft(u) u_hat *= self.symbol_v if is_v else self.symbol_u u_hat *= self.dealiasing_mask return self.ifft(u_hat) def parse_equation(self, equation)-
Parse the PDE to separate linear and nonlinear terms, symbolic operators (Op), source terms, and pseudo-differential operators (psiOp).
This method rewrites the input equation in standard form (lhs - rhs = 0), expands it, and classifies each term into one of the following categories:
- Linear terms involving derivatives or the unknown function u
- Nonlinear terms (products with u, powers of u, etc.)
- Symbolic pseudo-differential operators (Op)
- Source terms (independent of u)
- Pseudo-differential operators (psiOp)
Parameters equation (sympy.Eq): The partial differential equation to be analyzed. Can be provided as an Eq object or a sympy expression.
Returns
tuple-
A 5-tuple containing:
- linear_terms (dict): Mapping from derivative/function to coefficient.
- nonlinear_terms (list): List of terms classified as nonlinear.
- symbol_terms (list): List of (coefficient, symbolic operator) pairs.
- source_terms (list): List of terms independent of the unknown function.
- pseudo_terms (list): List of (coefficient, pseudo-differential symbol) pairs.
Notes
- If
psiOpis present in the equation, expansion is skipped for safety. - When
psiOpis used, only nonlinear terms, source terms, and possibly a time derivative are allowed; other linear terms and symbolic operators (Op) are forbidden. - Classification logic includes:
- Detection of nonlinear structures like products or powers of u
- Mixed terms involving both u and its derivatives
- External symbolic operators (Op) and pseudo-differential operators (psiOp)
Expand source code
def parse_equation(self, equation): """ Parse the PDE to separate linear and nonlinear terms, symbolic operators (Op), source terms, and pseudo-differential operators (psiOp). This method rewrites the input equation in standard form (lhs - rhs = 0), expands it, and classifies each term into one of the following categories: - Linear terms involving derivatives or the unknown function u - Nonlinear terms (products with u, powers of u, etc.) - Symbolic pseudo-differential operators (Op) - Source terms (independent of u) - Pseudo-differential operators (psiOp) Parameters equation (sympy.Eq): The partial differential equation to be analyzed. Can be provided as an Eq object or a sympy expression. Returns: tuple: A 5-tuple containing: - linear_terms (dict): Mapping from derivative/function to coefficient. - nonlinear_terms (list): List of terms classified as nonlinear. - symbol_terms (list): List of (coefficient, symbolic operator) pairs. - source_terms (list): List of terms independent of the unknown function. - pseudo_terms (list): List of (coefficient, pseudo-differential symbol) pairs. Notes: - If `psiOp` is present in the equation, expansion is skipped for safety. - When `psiOp` is used, only nonlinear terms, source terms, and possibly a time derivative are allowed; other linear terms and symbolic operators (Op) are forbidden. - Classification logic includes: - Detection of nonlinear structures like products or powers of u - Mixed terms involving both u and its derivatives - External symbolic operators (Op) and pseudo-differential operators (psiOp) """ def is_nonlinear_term(term, u_func): # If the term contains functions (Abs, sin, exp, ...) applied to u if term.has(u_func): for sub in preorder_traversal(term): if isinstance(sub, Function) and sub.has(u_func) and sub.func != u_func.func: return True # If the term contains a nonlinear power of u if term.has(Pow): for pow_term in term.atoms(Pow): if pow_term.base == u_func and pow_term.exp != 1: return True # If the term is a product containing u and its derivative if term.func == Mul: factors = term.args has_u = any((f.has(u_func) and not isinstance(f, Derivative) for f in factors)) has_derivative = any((isinstance(f, Derivative) and f.expr.func == u_func.func for f in factors)) if has_u and has_derivative: return True return False print("\n********************") print("* Equation parsing *") print("********************\n") if isinstance(equation, Eq): lhs = equation.lhs - equation.rhs else: lhs = equation print(f"\nEquation rewritten in standard form: {lhs}") if lhs.has(psiOp): print("⚠️ psiOp detected: skipping expansion for safety") lhs_expanded = lhs else: lhs_expanded = expand(lhs) print(f"\nExpanded equation: {lhs_expanded}") linear_terms = {} nonlinear_terms = [] symbol_terms = [] source_terms = [] pseudo_terms = [] for term in lhs_expanded.as_ordered_terms(): print(f"Analyzing term: {term}") if isinstance(term, psiOp): expr = term.args[0] pseudo_terms.append((1, expr)) print(" --> Classified as pseudo linear term (psiOp)") continue # Otherwise, look for psiOp inside (general case) if term.has(psiOp): psiops = term.atoms(psiOp) for psi in psiops: try: coeff = simplify(term / psi) expr = psi.args[0] pseudo_terms.append((coeff, expr)) print(" --> Classified as pseudo linear term (psiOp)") except Exception as e: print(f" ⚠️ Failed to extract psiOp coefficient in term: {term}") print(f" Reason: {e}") nonlinear_terms.append(term) print(" --> Fallback: classified as nonlinear") continue if term.has(Op): ops = term.atoms(Op) for op in ops: coeff = term / op expr = op.args[0] symbol_terms.append((coeff, expr)) print(" --> Classified as symbolic linear term (Op)") continue if is_nonlinear_term(term, self.u): nonlinear_terms.append(term) print(" --> Classified as nonlinear") continue derivs = term.atoms(Derivative) if derivs: deriv = derivs.pop() coeff = term / deriv linear_terms[deriv] = linear_terms.get(deriv, 0) + coeff print(f" Derivative found: {deriv}") print(" --> Classified as linear") elif self.u in term.atoms(Function): coeff = term.as_coefficients_dict().get(self.u, 1) linear_terms[self.u] = linear_terms.get(self.u, 0) + coeff print(" --> Classified as linear") else: source_terms.append(term) print(" --> Classified as source term") print(f"Final linear terms: {linear_terms}") print(f"Final nonlinear terms: {nonlinear_terms}") print(f"Symbol terms: {symbol_terms}") print(f"Pseudo terms: {pseudo_terms}") print(f"Source terms: {source_terms}") if pseudo_terms: # Check if a time derivative is present among the linear terms has_time_derivative = any( isinstance(term, Derivative) and self.t in [v for v, _ in term.variable_count] for term in linear_terms ) # Extract non-temporal linear terms invalid_linear_terms = { term: coeff for term, coeff in linear_terms.items() if not ( isinstance(term, Derivative) and self.t in [v for v, _ in term.variable_count] ) and term != self.u # exclusion of the simple u term (without derivative) } if invalid_linear_terms or symbol_terms: raise ValueError( "When psiOp is used, only nonlinear terms, source terms, " "and possibly a time derivative are allowed. " "Other linear terms and Ops are forbidden." ) return linear_terms, nonlinear_terms, symbol_terms, source_terms, pseudo_terms def plot_energy(self, log=False)-
Plot the time evolution of the total energy for wave equations. Visualizes the energy computed during simulation for both 1D and 2D cases. Requires temporal_order=2 and prior execution of compute_energy() during solve().
Parameters
log : bool If True, displays energy on a logarithmic scale to highlight exponential decay/growth.
Notes
- Energy is defined as E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹⸍²u|² ] dx
- Only available if energy monitoring was activated in solve()
- Automatically skips plotting if no energy data is available
Displays
- Time vs. Total Energy plot with grid and legend
- Appropriate axis labels and dimensional context (1D/2D)
- Logarithmic or linear scaling based on input parameter
Expand source code
def plot_energy(self, log=False): """ Plot the time evolution of the total energy for wave equations. Visualizes the energy computed during simulation for both 1D and 2D cases. Requires temporal_order=2 and prior execution of compute_energy() during solve(). Parameters: log : bool If True, displays energy on a logarithmic scale to highlight exponential decay/growth. Notes: - Energy is defined as E(t) = 1/2 ∫ [ (∂ₜu)² + |L¹⸍²u|² ] dx - Only available if energy monitoring was activated in solve() - Automatically skips plotting if no energy data is available Displays: - Time vs. Total Energy plot with grid and legend - Appropriate axis labels and dimensional context (1D/2D) - Logarithmic or linear scaling based on input parameter """ if not hasattr(self, 'energy_history') or not self.energy_history: print("No energy data recorded. Call compute_energy() within solve().") return # Time vector for plotting t = np.linspace(0, self.Lt, len(self.energy_history)) # Create the figure plt.figure(figsize=(6, 4)) if log: plt.semilogy(t, self.energy_history, label="Energy (log scale)") else: plt.plot(t, self.energy_history, label="Energy") # Axis labels and title plt.xlabel("Time") plt.ylabel("Total energy") plt.title("Energy evolution ({}D)".format(self.dim)) # Display options plt.grid(True) plt.legend() plt.tight_layout() plt.show() def plot_symbol(self, component='abs', k_range=None, cmap='viridis')-
Visualize the spectral symbol L(k) or L(kx, ky) in 1D or 2D.
This method plots the linear operator's symbolic Fourier representation either as a function of a single wavenumber k (1D), or two wavenumbers kx and ky (2D). The user can choose to display the real part, imaginary part, or absolute value of the symbol.
Parameters
component:str {'abs', 're', 'im'}- Component of the symbol to visualize:
- 'abs' : absolute value |a(k)| - 're' : real part Re[a(k)] - 'im' : imaginary part Im[a(k)] k_range:tuple (kmin, kmax, N), optional-
Wavenumber range for evaluation:
- kmin: minimum wavenumber - kmax: maximum wavenumber - N: number of sampling pointsIf None, defaults to [-10, 10] with high resolution.
cmap:str, optional- Colormap used for 2D surface plots. Default is 'viridis'.
Raises
ValueError: If the spatial dimension is not 1D or 2D.Notes
- In 1D, the symbol is plotted using a standard 2D line plot.
- In 2D, a 3D surface plot is generated with color-mapped height.
- Symbol evaluation uses self.L(k), which must be defined and callable.
Expand source code
def plot_symbol(self, component="abs", k_range=None, cmap="viridis"): """ Visualize the spectral symbol L(k) or L(kx, ky) in 1D or 2D. This method plots the linear operator's symbolic Fourier representation either as a function of a single wavenumber k (1D), or two wavenumbers kx and ky (2D). The user can choose to display the real part, imaginary part, or absolute value of the symbol. Parameters ---------- component : str {'abs', 're', 'im'} Component of the symbol to visualize: - 'abs' : absolute value |a(k)| - 're' : real part Re[a(k)] - 'im' : imaginary part Im[a(k)] k_range : tuple (kmin, kmax, N), optional Wavenumber range for evaluation: - kmin: minimum wavenumber - kmax: maximum wavenumber - N: number of sampling points If None, defaults to [-10, 10] with high resolution. cmap : str, optional Colormap used for 2D surface plots. Default is 'viridis'. Raises ------ ValueError: If the spatial dimension is not 1D or 2D. Notes: - In 1D, the symbol is plotted using a standard 2D line plot. - In 2D, a 3D surface plot is generated with color-mapped height. - Symbol evaluation uses self.L(k), which must be defined and callable. """ print("\n*******************") print("* Symbol plotting *") print("*******************\n") assert component in ("abs", "re", "im"), "component must be 'abs', 're' or 'im'" if self.dim == 1: if k_range is None: k_vals = np.linspace(-10, 10, 1000) else: kmin, kmax, N = k_range k_vals = np.linspace(kmin, kmax, N) L_vals = self.L(k_vals) if component == "re": vals = np.real(L_vals) label = "Re[a(k)]" elif component == "im": vals = np.imag(L_vals) label = "Im[a(k)]" else: vals = np.abs(L_vals) label = "|a(k)|" plt.plot(k_vals, vals) plt.xlabel("k") plt.ylabel(label) plt.title(f"Spectral symbol: {label}") plt.grid(True) plt.show() elif self.dim == 2: if k_range is None: k_vals = np.linspace(-10, 10, 300) else: kmin, kmax, N = k_range k_vals = np.linspace(kmin, kmax, N) KX, KY = np.meshgrid(k_vals, k_vals) L_vals = self.L(KX, KY) if component == "re": Z = np.real(L_vals) title = "Re[a(kx, ky)]" elif component == "im": Z = np.imag(L_vals) title = "Im[a(kx, ky)]" else: Z = np.abs(L_vals) title = "|a(kx, ky)|" fig = plt.figure(figsize=(8, 6)) ax = fig.add_subplot(111, projection='3d') surf = ax.plot_surface(KX, KY, Z, cmap=cmap, edgecolor='none', antialiased=True) fig.colorbar(surf, ax=ax, shrink=0.6) ax.set_xlabel("kx") ax.set_ylabel("ky") ax.set_zlabel(title) ax.set_title(f"2D spectral symbol: {title}") plt.tight_layout() plt.show() else: raise ValueError("Only 1D and 2D supported.") def prepare_symbol_tables(self)-
Precompute and store evaluated pseudo-differential operator symbols for spectral methods.
This method evaluates all pseudo-differential operators (ψOp) present in the PDE over the spatial and frequency grids, scales them by their respective coefficients, and combines them into a single composite symbol used in time-stepping and inversion.
The evaluation is performed via the
evaluatemethod of each PseudoDifferentialOperator, which computes p(x, ξ) or p(x, y, ξ, η) numerically over the current grid configuration.Side Effects: self.precomputed_symbols : list of (coeff, symbol_array) Each tuple contains a coefficient and its evaluated symbol on the grid. self.combined_symbol : np.ndarray Sum of all scaled symbol arrays: ∑(coeffₖ * ψₖ(x, ξ))
Raises
ValueError- If the spatial dimension is not 1D or 2D.
Expand source code
def prepare_symbol_tables(self): """ Precompute and store evaluated pseudo-differential operator symbols for spectral methods. This method evaluates all pseudo-differential operators (ψOp) present in the PDE over the spatial and frequency grids, scales them by their respective coefficients, and combines them into a single composite symbol used in time-stepping and inversion. The evaluation is performed via the `evaluate` method of each PseudoDifferentialOperator, which computes p(x, ξ) or p(x, y, ξ, η) numerically over the current grid configuration. Side Effects: self.precomputed_symbols : list of (coeff, symbol_array) Each tuple contains a coefficient and its evaluated symbol on the grid. self.combined_symbol : np.ndarray Sum of all scaled symbol arrays: ∑(coeffₖ * ψₖ(x, ξ)) Raises: ValueError: If the spatial dimension is not 1D or 2D. """ self.precomputed_symbols = [] self.combined_symbol = 0 for coeff, psi in self.psi_ops: if self.dim == 1: raw = psi.evaluate(self.X, None, self.KX, None) elif self.dim == 2: raw = psi.evaluate(self.X, self.Y, self.KX, self.KY) else: raise ValueError('Unsupported spatial dimension.') raw_flat = raw.flatten() converted = np.array([complex(N(val)) for val in raw_flat], dtype=np.complex128) raw_eval = converted.reshape(raw.shape) self.precomputed_symbols.append((coeff, raw_eval)) self.combined_symbol = sum((coeff * sym for coeff, sym in self.precomputed_symbols)) self.combined_symbol = np.array(self.combined_symbol, dtype=np.complex128) def setup(self, Lx, Ly=None, Nx=None, Ny=None, Lt=1.0, Nt=100, boundary_condition='periodic', initial_condition=None, initial_velocity=None, n_frames=100)-
Configure the spatial/temporal grid and initialize the solution field.
This method sets up the computational domain, initializes spatial and temporal grids, applies boundary conditions, and prepares symbolic and numerical operators. It also performs essential analyses such as:
- CFL condition verification (for stability) - Symbol analysis (e.g., dispersion relation, regularity) - Wave propagation analysis for second-order equationsIf pseudo-differential operators (ψOp) are present, symbolic analysis is skipped in favor of interactive exploration via
interactive_symbol_analysis.Parameters
Lx:float- Size of the spatial domain along x-axis.
Ly:float, optional- Size of the spatial domain along y-axis (for 2D problems).
Nx:int- Number of spatial points along x-axis.
Ny:int, optional- Number of spatial points along y-axis (for 2D problems).
Lt:float, default=1.0- Total simulation time.
Nt:int, default=100- Number of time steps.
initial_condition:callable- Function returning the initial state u(x, 0) or u(x, y, 0).
initial_velocity:callable, optional- Function returning the initial time derivative ∂ₜu(x, 0) or ∂ₜu(x, y, 0), required for second-order equations.
n_frames:int, default=100- Number of time frames to store during simulation for visualization or output.
Raises
ValueError- If mandatory parameters are missing (e.g., Nx not given in 1D, Ly/Ny not given in 2D).
Notes
- The spatial discretization assumes periodic boundary conditions by default.
- Fourier transforms are computed using real-to-complex FFTs (
scipy.fft.fft,fft2). - Frequency arrays (
KX,KY) are defined following standard spectral conventions. - Dealiasing is applied using a sharp cutoff filter at a fraction of the maximum frequency.
- For second-order equations, initial acceleration is derived from the governing operator.
- Symbolic analysis includes plotting of the symbol's real/imaginary/absolute values, wavefront propagation, and dispersion relation.
See Also
setup_1D- Sets up internal variables for one-dimensional problems.
setup_2D- Sets up internal variables for two-dimensional problems.
initialize_conditions- Applies initial data and enforces compatibility.
check_cfl_condition- Verifies time step against stability constraints.
plot_symbol- Visualizes the linear operator’s symbol in frequency space.
analyze_wave_propagation- Analyzes group velocity and wavefront dynamics.
interactive_symbol_analysis- Interactive tools for ψOp-based equations.
Expand source code
def setup(self, Lx, Ly=None, Nx=None, Ny=None, Lt=1.0, Nt=100, boundary_condition='periodic', initial_condition=None, initial_velocity=None, n_frames=100): """ Configure the spatial/temporal grid and initialize the solution field. This method sets up the computational domain, initializes spatial and temporal grids, applies boundary conditions, and prepares symbolic and numerical operators. It also performs essential analyses such as: - CFL condition verification (for stability) - Symbol analysis (e.g., dispersion relation, regularity) - Wave propagation analysis for second-order equations If pseudo-differential operators (ψOp) are present, symbolic analysis is skipped in favor of interactive exploration via `interactive_symbol_analysis`. Parameters ---------- Lx : float Size of the spatial domain along x-axis. Ly : float, optional Size of the spatial domain along y-axis (for 2D problems). Nx : int Number of spatial points along x-axis. Ny : int, optional Number of spatial points along y-axis (for 2D problems). Lt : float, default=1.0 Total simulation time. Nt : int, default=100 Number of time steps. initial_condition : callable Function returning the initial state u(x, 0) or u(x, y, 0). initial_velocity : callable, optional Function returning the initial time derivative ∂ₜu(x, 0) or ∂ₜu(x, y, 0), required for second-order equations. n_frames : int, default=100 Number of time frames to store during simulation for visualization or output. Raises ------ ValueError If mandatory parameters are missing (e.g., Nx not given in 1D, Ly/Ny not given in 2D). Notes ----- - The spatial discretization assumes periodic boundary conditions by default. - Fourier transforms are computed using real-to-complex FFTs (`scipy.fft.fft`, `fft2`). - Frequency arrays (`KX`, `KY`) are defined following standard spectral conventions. - Dealiasing is applied using a sharp cutoff filter at a fraction of the maximum frequency. - For second-order equations, initial acceleration is derived from the governing operator. - Symbolic analysis includes plotting of the symbol's real/imaginary/absolute values, wavefront propagation, and dispersion relation. See Also -------- setup_1D : Sets up internal variables for one-dimensional problems. setup_2D : Sets up internal variables for two-dimensional problems. initialize_conditions : Applies initial data and enforces compatibility. check_cfl_condition : Verifies time step against stability constraints. plot_symbol : Visualizes the linear operator’s symbol in frequency space. analyze_wave_propagation : Analyzes group velocity and wavefront dynamics. interactive_symbol_analysis : Interactive tools for ψOp-based equations. """ # Temporal parameters self.Lt, self.Nt = Lt, Nt self.dt = Lt / Nt self.n_frames = n_frames self.frames = [] self.initial_condition = initial_condition self.boundary_condition = boundary_condition if self.boundary_condition == 'dirichlet' and not self.has_psi: raise ValueError( "Dirichlet boundary conditions require the equation to be defined via a pseudo-differential operator (psiOp). " "Please provide an equation involving psiOp for non-periodic boundary treatment." ) # Dimension checks if self.dim == 1: if Nx is None: raise ValueError("Nx must be specified in 1D.") self.setup_1D(Lx, Nx) else: if None in (Ly, Ny): raise ValueError("In 2D, Ly and Ny must be provided.") self.setup_2D(Lx, Ly, Nx, Ny) # Initialization of solution and velocities if not self.is_stationary: self.initialize_conditions(initial_condition, initial_velocity) # Symbol analysis if present if self.has_psi: print("⚠️ For psiOp, use interactive_symbol_analysis.") else: if self.L_symbolic == 0: print("⚠️ Linear operator is null.") else: self.check_cfl_condition() self.check_symbol_conditions() self.plot_symbol() if self.temporal_order == 2: self.analyze_wave_propagation() def setup_1D(self, Lx, Nx)-
Configure internal variables for one-dimensional (1D) problems.
This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution.
It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2].
Parameters
Lx:float- Physical size of the spatial domain along the x-axis.
Nx:int- Number of grid points in the x-direction.
Attributes Set
- self.Lx : float Size of the spatial domain.
- self.Nx : int Number of spatial points.
- self.x_grid : np.ndarray 1D array of spatial coordinates.
- self.X : np.ndarray
Alias to
self.x_grid, used in physical space computations. - self.kx : np.ndarray Array of wavenumbers corresponding to the Fourier transform.
- self.KX : np.ndarray
Alias to
self.kx, used in frequency space computations. - self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations.
- self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(k) · dt).
- self.omega_val : np.ndarray Frequency values ω(k) = Re[√(L(k))] used in second-order time stepping.
- self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(k)·dt for dispersive propagation.
- self.inv_omega : np.ndarray Inverse of ω(k), used to avoid division-by-zero in time stepping.
Notes
- Frequencies are computed using
scipy.fft.fftfreqand then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on
self.dealiasing_ratio. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via
prepare_symbol_tables. - For second-order equations, the dispersion relation ω(k) is extracted from the linear operator L(k).
See Also
setup_2D- Equivalent setup for two-dimensional problems.
prepare_symbol_tables- Precomputes symbolic arrays for ψOp evaluation.
setup_omega_terms- Sets up terms involving ω(k) for second-order evolution.
Expand source code
def setup_1D(self, Lx, Nx): """ Configure internal variables for one-dimensional (1D) problems. This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution. It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2]. Parameters ---------- Lx : float Physical size of the spatial domain along the x-axis. Nx : int Number of grid points in the x-direction. Attributes Set -------------- - self.Lx : float Size of the spatial domain. - self.Nx : int Number of spatial points. - self.x_grid : np.ndarray 1D array of spatial coordinates. - self.X : np.ndarray Alias to `self.x_grid`, used in physical space computations. - self.kx : np.ndarray Array of wavenumbers corresponding to the Fourier transform. - self.KX : np.ndarray Alias to `self.kx`, used in frequency space computations. - self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations. - self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(k) · dt). - self.omega_val : np.ndarray Frequency values ω(k) = Re[√(L(k))] used in second-order time stepping. - self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(k)·dt for dispersive propagation. - self.inv_omega : np.ndarray Inverse of ω(k), used to avoid division-by-zero in time stepping. Notes ----- - Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`. - For second-order equations, the dispersion relation ω(k) is extracted from the linear operator L(k). See Also -------- setup_2D : Equivalent setup for two-dimensional problems. prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation. setup_omega_terms : Sets up terms involving ω(k) for second-order evolution. """ self.Lx, self.Nx = Lx, Nx self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False) self.X = self.x_grid self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx) self.KX = self.kx # Dealiasing mask k_max = self.dealiasing_ratio * np.max(np.abs(self.kx)) self.dealiasing_mask = (np.abs(self.KX) <= k_max) # Preparation of symbol or linear operator if self.has_psi: self.prepare_symbol_tables() else: L_vals = np.array(self.L(self.KX), dtype=np.complex128) self.exp_L = np.exp(L_vals * self.dt) if self.temporal_order == 2: omega_val = self.omega(self.KX) self.setup_omega_terms(omega_val) def setup_2D(self, Lx, Ly, Nx, Ny)-
Configure internal variables for two-dimensional (2D) problems.
This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution.
It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2] × [-Ly/2, Ly/2].
Parameters
Lx:float- Physical size of the spatial domain along the x-axis.
Ly:float- Physical size of the spatial domain along the y-axis.
Nx:int- Number of grid points along the x-direction.
Ny:int- Number of grid points along the y-direction.
Attributes Set
- self.Lx, self.Ly : float Size of the spatial domain in each direction.
- self.Nx, self.Ny : int Number of spatial points in each direction.
- self.x_grid, self.y_grid : np.ndarray 1D arrays of spatial coordinates in x and y directions.
- self.X, self.Y : np.ndarray 2D meshgrids of spatial coordinates for physical space computations.
- self.kx, self.ky : np.ndarray Arrays of wavenumbers corresponding to Fourier transforms in x and y directions.
- self.KX, self.KY : np.ndarray Meshgrids of wavenumbers used in frequency space computations.
- self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations.
- self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(kx, ky) · dt).
- self.omega_val : np.ndarray Frequency values ω(kx, ky) = Re[√(L(kx, ky))] used in second-order time stepping.
- self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(kx, ky)·dt for dispersive propagation.
- self.inv_omega : np.ndarray Inverse of ω(kx, ky), used to avoid division-by-zero in time stepping.
Notes
- Frequencies are computed using
scipy.fft.fftfreqand then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on
self.dealiasing_ratio. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via
prepare_symbol_tables. - For second-order equations, the dispersion relation ω(kx, ky) is extracted from the linear operator L(kx, ky).
See Also
setup_1D- Equivalent setup for one-dimensional problems.
prepare_symbol_tables- Precomputes symbolic arrays for ψOp evaluation.
setup_omega_terms- Sets up terms involving ω(kx, ky) for second-order evolution.
Expand source code
def setup_2D(self, Lx, Ly, Nx, Ny): """ Configure internal variables for two-dimensional (2D) problems. This private method initializes spatial and frequency grids, applies dealiasing, and prepares either pseudo-differential symbols or linear operators for use in time evolution. It assumes periodic boundary conditions and uses real-to-complex FFT conventions. The spatial domain is centered at zero: [-Lx/2, Lx/2] × [-Ly/2, Ly/2]. Parameters ---------- Lx : float Physical size of the spatial domain along the x-axis. Ly : float Physical size of the spatial domain along the y-axis. Nx : int Number of grid points along the x-direction. Ny : int Number of grid points along the y-direction. Attributes Set -------------- - self.Lx, self.Ly : float Size of the spatial domain in each direction. - self.Nx, self.Ny : int Number of spatial points in each direction. - self.x_grid, self.y_grid : np.ndarray 1D arrays of spatial coordinates in x and y directions. - self.X, self.Y : np.ndarray 2D meshgrids of spatial coordinates for physical space computations. - self.kx, self.ky : np.ndarray Arrays of wavenumbers corresponding to Fourier transforms in x and y directions. - self.KX, self.KY : np.ndarray Meshgrids of wavenumbers used in frequency space computations. - self.dealiasing_mask : np.ndarray Boolean mask used to suppress aliased frequencies during nonlinear calculations. - self.exp_L : np.ndarray Exponential of the linear operator scaled by time step: exp(L(kx, ky) · dt). - self.omega_val : np.ndarray Frequency values ω(kx, ky) = Re[√(L(kx, ky))] used in second-order time stepping. - self.cos_omega_dt, self.sin_omega_dt : np.ndarray Cosine and sine of ω(kx, ky)·dt for dispersive propagation. - self.inv_omega : np.ndarray Inverse of ω(kx, ky), used to avoid division-by-zero in time stepping. Notes ----- - Frequencies are computed using `scipy.fft.fftfreq` and then shifted to center zero frequency. - Dealiasing is applied using a sharp cutoff filter based on `self.dealiasing_ratio`. - If pseudo-differential operators (ψOp) are present, symbolic tables are precomputed via `prepare_symbol_tables`. - For second-order equations, the dispersion relation ω(kx, ky) is extracted from the linear operator L(kx, ky). See Also -------- setup_1D : Equivalent setup for one-dimensional problems. prepare_symbol_tables : Precomputes symbolic arrays for ψOp evaluation. setup_omega_terms : Sets up terms involving ω(kx, ky) for second-order evolution. """ self.Lx, self.Ly = Lx, Ly self.Nx, self.Ny = Nx, Ny self.x_grid = np.linspace(-Lx/2, Lx/2, Nx, endpoint=False) self.y_grid = np.linspace(-Ly/2, Ly/2, Ny, endpoint=False) self.X, self.Y = np.meshgrid(self.x_grid, self.y_grid, indexing='ij') self.kx = 2 * np.pi * fftfreq(Nx, d=Lx / Nx) self.ky = 2 * np.pi * fftfreq(Ny, d=Ly / Ny) self.KX, self.KY = np.meshgrid(self.kx, self.ky, indexing='ij') # Dealiasing mask kx_max = self.dealiasing_ratio * np.max(np.abs(self.kx)) ky_max = self.dealiasing_ratio * np.max(np.abs(self.ky)) self.dealiasing_mask = (np.abs(self.KX) <= kx_max) & (np.abs(self.KY) <= ky_max) # Preparation of symbol or linear operator if self.has_psi: self.prepare_symbol_tables() else: L_vals = self.L(self.KX, self.KY) self.exp_L = np.exp(L_vals * self.dt) if self.temporal_order == 2: omega_val = self.omega(self.KX, self.KY) self.setup_omega_terms(omega_val) def setup_omega_terms(self, omega_val)-
Initialize terms derived from the angular frequency ω for time evolution.
This private method precomputes and stores key trigonometric and inverse quantities based on the dispersion relation ω(k), used in second-order time integration schemes.
These values are essential for solving wave-like equations with dispersive behavior: cos(ω·dt), sin(ω·dt), 1/ω
The inverse frequency is computed safely to avoid division by zero.
Parameters
omega_val:np.ndarray- Array of angular frequency values ω(k) evaluated at discrete wavenumbers. Can be one-dimensional (1D) or two-dimensional (2D) depending on spatial dimension.
Attributes Set
- self.omega_val : np.ndarray Copy of the input angular frequency array.
- self.cos_omega_dt : np.ndarray Cosine of ω(k) multiplied by time step: cos(ω(k) · dt).
- self.sin_omega_dt : np.ndarray Sine of ω(k) multiplied by time step: sin(ω(k) · dt).
- self.inv_omega : np.ndarray Inverse of ω(k), with zeros where ω(k) == 0 to avoid division by zero.
Notes
- This method is typically called during setup when solving second-order PDEs involving dispersive waves (e.g., Klein-Gordon, Schrödinger, or water wave equations).
- The safe computation of 1/ω ensures numerical stability even when low frequencies are present.
- These precomputed arrays are used in spectral propagators for accurate time stepping.
See Also
setup_1D- Sets up internal variables for one-dimensional problems.
setup_2D- Sets up internal variables for two-dimensional problems.
solve- Time integration using the computed frequency terms.
Expand source code
def setup_omega_terms(self, omega_val): """ Initialize terms derived from the angular frequency ω for time evolution. This private method precomputes and stores key trigonometric and inverse quantities based on the dispersion relation ω(k), used in second-order time integration schemes. These values are essential for solving wave-like equations with dispersive behavior: cos(ω·dt), sin(ω·dt), 1/ω The inverse frequency is computed safely to avoid division by zero. Parameters ---------- omega_val : np.ndarray Array of angular frequency values ω(k) evaluated at discrete wavenumbers. Can be one-dimensional (1D) or two-dimensional (2D) depending on spatial dimension. Attributes Set -------------- - self.omega_val : np.ndarray Copy of the input angular frequency array. - self.cos_omega_dt : np.ndarray Cosine of ω(k) multiplied by time step: cos(ω(k) · dt). - self.sin_omega_dt : np.ndarray Sine of ω(k) multiplied by time step: sin(ω(k) · dt). - self.inv_omega : np.ndarray Inverse of ω(k), with zeros where ω(k) == 0 to avoid division by zero. Notes ----- - This method is typically called during setup when solving second-order PDEs involving dispersive waves (e.g., Klein-Gordon, Schrödinger, or water wave equations). - The safe computation of 1/ω ensures numerical stability even when low frequencies are present. - These precomputed arrays are used in spectral propagators for accurate time stepping. See Also -------- setup_1D : Sets up internal variables for one-dimensional problems. setup_2D : Sets up internal variables for two-dimensional problems. solve : Time integration using the computed frequency terms. """ self.omega_val = omega_val self.cos_omega_dt = np.cos(omega_val * self.dt) self.sin_omega_dt = np.sin(omega_val * self.dt) self.inv_omega = np.zeros_like(omega_val) nonzero = omega_val != 0 self.inv_omega[nonzero] = 1.0 / omega_val[nonzero] def show_stationary_solution(self, u=None, component='abs', cmap='viridis')-
Display the stationary solution computed by solve_stationary_psiOp.
This method visualizes the solution of a pseudo-differential equation solved in stationary mode. It supports both 1D and 2D spatial domains, with options to display different components of the solution (real, imaginary, absolute value, or phase).
Parameters
u:ndarray, optional- Precomputed solution array. If None, calls solve_stationary_psiOp() to compute the solution.
component:str, optional{'real', 'imag', 'abs', 'angle'}- Component of the complex-valued solution to display: - 'real': Real part - 'imag': Imaginary part - 'abs' : Absolute value (modulus) - 'angle' : Phase (argument)
cmap:str, optional- Colormap used for 2D visualization (default: 'viridis').
Raises
ValueError- If an invalid component is specified or if the spatial dimension is not supported (only 1D and 2D are implemented).
Notes
- In 1D, the solution is displayed using a standard line plot.
- In 2D, the solution is visualized as a 3D surface plot.
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def show_stationary_solution(self, u=None, component='abs', cmap='viridis'): """ Display the stationary solution computed by solve_stationary_psiOp. This method visualizes the solution of a pseudo-differential equation solved in stationary mode. It supports both 1D and 2D spatial domains, with options to display different components of the solution (real, imaginary, absolute value, or phase). Parameters ---------- u : ndarray, optional Precomputed solution array. If None, calls solve_stationary_psiOp() to compute the solution. component : str, optional {'real', 'imag', 'abs', 'angle'} Component of the complex-valued solution to display: - 'real': Real part - 'imag': Imaginary part - 'abs' : Absolute value (modulus) - 'angle' : Phase (argument) cmap : str, optional Colormap used for 2D visualization (default: 'viridis'). Raises ------ ValueError If an invalid component is specified or if the spatial dimension is not supported (only 1D and 2D are implemented). Notes ----- - In 1D, the solution is displayed using a standard line plot. - In 2D, the solution is visualized as a 3D surface plot. """ def get_component(u): if component == 'real': return np.real(u) elif component == 'imag': return np.imag(u) elif component == 'abs': return np.abs(u) elif component == 'angle': return np.angle(u) else: raise ValueError("Invalid component") if u is None: u = self.solve_stationary_psiOp() if self.dim == 1: # Plot the solution in 1D plt.figure(figsize=(8, 4)) plt.plot(self.x_grid, get_component(u), label=f'{component} of u') plt.xlabel('x') plt.ylabel(f'{component} of u') plt.title('Stationary solution (1D)') plt.grid(True) plt.legend() plt.tight_layout() plt.show() elif self.dim == 2: fig = plt.figure(figsize=(12, 6)) ax = fig.add_subplot(111, projection='3d') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel(f'{component.title()} of u') plt.title('Stationary solution (2D)') data0 = get_component(u) ax.plot_surface(self.X, self.Y, data0, cmap='viridis') plt.tight_layout() plt.show() else: raise ValueError("Only 1D and 2D display are supported.") def solve(self)-
Solve the partial differential equation numerically using spectral methods.
This method evolves the solution in time using a combination of: - Fourier-based linear evolution (with dealiasing) - Nonlinear term handling via pseudo-spectral evaluation - Support for pseudo-differential operators (psiOp) - Source terms and boundary conditions
The solver supports: - 1D and 2D spatial domains - First and second-order time evolution - Periodic and Dirichlet boundary conditions - Time-stepping schemes: default, ETD-RK4
Returns
list[np.ndarray]- A list of solution arrays at each saved time frame.
Side Effects: - Updates self.frames: stores solution snapshots - Updates self.energy_history: records total energy if enabled
Algorithm Overview: For each time step: 1. Evaluate source contributions (if any) 2. Apply time evolution: - Order 1: - With psiOp: uses step_order1_with_psi - With ETD-RK4: exponential time differencing - Default: linear + nonlinear update - Order 2: - With psiOp: uses step_order2_with_psi - With ETD-RK4: second-order exponential scheme - Default: second-order leapfrog-style update 3. Enforce boundary conditions 4. Save solution snapshot periodically 5. Record energy (for second-order systems without psiOp)
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def solve(self): """ Solve the partial differential equation numerically using spectral methods. This method evolves the solution in time using a combination of: - Fourier-based linear evolution (with dealiasing) - Nonlinear term handling via pseudo-spectral evaluation - Support for pseudo-differential operators (psiOp) - Source terms and boundary conditions The solver supports: - 1D and 2D spatial domains - First and second-order time evolution - Periodic and Dirichlet boundary conditions - Time-stepping schemes: default, ETD-RK4 Returns: list[np.ndarray]: A list of solution arrays at each saved time frame. Side Effects: - Updates self.frames: stores solution snapshots - Updates self.energy_history: records total energy if enabled Algorithm Overview: For each time step: 1. Evaluate source contributions (if any) 2. Apply time evolution: - Order 1: - With psiOp: uses step_order1_with_psi - With ETD-RK4: exponential time differencing - Default: linear + nonlinear update - Order 2: - With psiOp: uses step_order2_with_psi - With ETD-RK4: second-order exponential scheme - Default: second-order leapfrog-style update 3. Enforce boundary conditions 4. Save solution snapshot periodically 5. Record energy (for second-order systems without psiOp) """ print('\n*******************') print('* Solving the PDE *') print('*******************\n') save_interval = max(1, self.Nt // self.n_frames) self.energy_history = [] for step in range(self.Nt): if hasattr(self, 'source_terms') and self.source_terms: source_contribution = np.zeros_like(self.X, dtype=np.float64) for term in self.source_terms: try: if self.dim == 1: source_func = lambdify((self.t, self.x), term, 'numpy') source_contribution += source_func(step * self.dt, self.X) elif self.dim == 2: source_func = lambdify((self.t, self.x, self.y), term, 'numpy') source_contribution += source_func(step * self.dt, self.X, self.Y) except Exception as e: print(f'Error evaluating source term {term}: {e}') else: source_contribution = 0 if self.temporal_order == 1: if self.has_psi: u_new = self.step_order1_with_psi(source_contribution) elif hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4': u_new = self.step_ETD_RK4(self.u_prev) else: u_hat = self.fft(self.u_prev) u_hat *= self.exp_L u_hat *= self.dealiasing_mask u_lin = self.ifft(u_hat) u_nl = self.apply_nonlinear(u_lin) u_new = u_lin + u_nl + source_contribution self.apply_boundary(u_new) self.u_prev = u_new elif self.temporal_order == 2: if self.has_psi: u_new = self.step_order2_with_psi(source_contribution) else: if hasattr(self, 'time_scheme') and self.time_scheme == 'ETD-RK4': u_new, v_new = self.step_ETD_RK4_order2(self.u_prev, self.v_prev) else: u_hat = self.fft(self.u_prev) v_hat = self.fft(self.v_prev) u_new_hat = self.cos_omega_dt * u_hat + self.sin_omega_dt * self.inv_omega * v_hat v_new_hat = -self.omega_val * self.sin_omega_dt * u_hat + self.cos_omega_dt * v_hat u_new = self.ifft(u_new_hat) v_new = self.ifft(v_new_hat) u_nl = self.apply_nonlinear(self.u_prev, is_v=False) v_nl = self.apply_nonlinear(self.v_prev, is_v=True) u_new += (u_nl + source_contribution) * self.dt ** 2 / 2 v_new += (u_nl + source_contribution) * self.dt self.apply_boundary(u_new) self.apply_boundary(v_new) self.u_prev = u_new self.v_prev = v_new if step % save_interval == 0: self.frames.append(self.u_prev.copy()) if self.temporal_order == 2 and (not self.has_psi): E = self.compute_energy() self.energy_history.append(E) return self.frames def solve_stationary_psiOp(self, order=3)-
Solve stationary pseudo-differential equations of the form P[u] = f(x) or P[u] = f(x,y) using asymptotic inversion.
This method computes the solution to a stationary (time-independent) pseudo-differential equation where the operator P is defined via symbolic expressions (psiOp). It constructs an asymptotic right inverse R such that P∘R ≈ Id, then applies it to the source term f using either direct Fourier multiplication (when the symbol is spatially independent) or Kohn–Nirenberg quantization (when spatial dependence is present).
The inversion is based on the principal symbol of the operator and its asymptotic expansion up to the given order. Ellipticity of the symbol is checked numerically before inversion to ensure well-posedness.
Parameters
order:int, default=3- Order of the asymptotic expansion used to construct the right inverse of the pseudo-differential operator.
method:str, optional- Inversion strategy: - 'diagonal' (default): Fast approximate inversion using diagonal operators in frequency space. - 'full' : Pointwise exact inversion (slower but more accurate).
Returns
ndarray- The computed solution u(x) in 1D or u(x, y) in 2D as a NumPy array over the spatial grid.
Raises
ValueError- If no pseudo-differential operator (psiOp) is defined. If linear or nonlinear terms other than psiOp are present. If the symbol is not elliptic on the grid. If no source term is provided for the right-hand side.
Notes
- The method assumes the problem is fully stationary: time derivatives must be absent.
- Requires the equation to be purely pseudo-differential (no Op, Derivative, or nonlinear terms).
- Symbol evaluation and inversion are dimension-aware (supports both 1D and 2D problems).
- Supports optimization paths when the symbol does not depend on spatial variables.
See Also
right_inverse_asymptotic- Constructs the asymptotic inverse of the pseudo-differential operator.
kohn_nirenberg : Numerical implementation of general pseudo-differential operators.is_elliptic_numerically : Verifies numerical ellipticity of the symbol.
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def solve_stationary_psiOp(self, order=3): """ Solve stationary pseudo-differential equations of the form P[u] = f(x) or P[u] = f(x,y) using asymptotic inversion. This method computes the solution to a stationary (time-independent) pseudo-differential equation where the operator P is defined via symbolic expressions (psiOp). It constructs an asymptotic right inverse R such that P∘R ≈ Id, then applies it to the source term f using either direct Fourier multiplication (when the symbol is spatially independent) or Kohn–Nirenberg quantization (when spatial dependence is present). The inversion is based on the principal symbol of the operator and its asymptotic expansion up to the given order. Ellipticity of the symbol is checked numerically before inversion to ensure well-posedness. Parameters ---------- order : int, default=3 Order of the asymptotic expansion used to construct the right inverse of the pseudo-differential operator. method : str, optional Inversion strategy: - 'diagonal' (default): Fast approximate inversion using diagonal operators in frequency space. - 'full' : Pointwise exact inversion (slower but more accurate). Returns ------- ndarray The computed solution u(x) in 1D or u(x, y) in 2D as a NumPy array over the spatial grid. Raises ------ ValueError If no pseudo-differential operator (psiOp) is defined. If linear or nonlinear terms other than psiOp are present. If the symbol is not elliptic on the grid. If no source term is provided for the right-hand side. Notes ----- - The method assumes the problem is fully stationary: time derivatives must be absent. - Requires the equation to be purely pseudo-differential (no Op, Derivative, or nonlinear terms). - Symbol evaluation and inversion are dimension-aware (supports both 1D and 2D problems). - Supports optimization paths when the symbol does not depend on spatial variables. See Also -------- right_inverse_asymptotic : Constructs the asymptotic inverse of the pseudo-differential operator. kohn_nirenberg : Numerical implementation of general pseudo-differential operators. is_elliptic_numerically : Verifies numerical ellipticity of the symbol. """ print("\n*******************************") print("* Solving the stationnary PDE *") print("*******************************\n") print("boundary condition: ",self.boundary_condition) if not self.has_psi: raise ValueError("Only supports problems with psiOp.") if self.linear_terms or self.nonlinear_terms: raise ValueError("Stationary psiOp problems must be linear and purely pseudo-differential.") if self.boundary_condition not in ('periodic', 'dirichlet'): raise ValueError( "For stationary PDEs, boundary conditions must be explicitly defined. " "Supported types are 'periodic' and 'dirichlet'." ) if self.dim == 1: x = self.x xi = symbols('xi', real=True) spatial_vars = (x,) freq_vars = (xi,) X, KX = self.X, self.KX elif self.dim == 2: x, y = self.x, self.y xi, eta = symbols('xi eta', real=True) spatial_vars = (x, y) freq_vars = (xi, eta) X, Y, KX, KY = self.X, self.Y, self.KX, self.KY else: raise ValueError("Unsupported spatial dimension.") total_symbol = sum(coeff * psi.expr for coeff, psi in self.psi_ops) psi_total = PseudoDifferentialOperator(total_symbol, spatial_vars, mode='symbol') # Check ellipticity if self.dim == 1: is_elliptic = psi_total.is_elliptic_numerically(X, KX) else: is_elliptic = psi_total.is_elliptic_numerically((X[:, 0], Y[0, :]), (KX[:, 0], KY[0, :])) if not is_elliptic: raise ValueError("❌ The pseudo-differential symbol is not numerically elliptic on the grid.") print("✅ Elliptic pseudo-differential symbol: inversion allowed.") R_symbol = psi_total.right_inverse_asymptotic(order=order) print("Right inverse asymptotic symbol:") pprint(R_symbol, num_columns=150) if self.dim == 1: if R_symbol.has(x): R_func = lambdify((x, xi), R_symbol, modules='numpy') else: R_func = lambdify((xi,), R_symbol, modules='numpy') else: if R_symbol.has(x) or R_symbol.has(y): R_func = lambdify((x, y, xi, eta), R_symbol, modules='numpy') else: R_func = lambdify((xi, eta), R_symbol, modules='numpy') # Build rhs if self.source_terms: f_expr = sum(self.source_terms) used_vars = [v for v in spatial_vars if f_expr.has(v)] f_func = lambdify(used_vars, -f_expr, modules='numpy') if self.dim == 1: rhs = f_func(self.x_grid) if used_vars else np.zeros_like(self.x_grid) else: rhs = f_func(self.X, self.Y) if used_vars else np.zeros_like(self.X) elif self.initial_condition: raise ValueError("Initial condition should be None for stationnary equation.") else: raise ValueError("No source term provided to construct the right-hand side.") f_hat = self.fft(rhs) if self.boundary_condition == 'periodic': if self.dim == 1: if not R_symbol.has(x): print("⚡ Optimization: symbol independent of x — direct product in Fourier.") R_vals = R_func(self.KX) u_hat = R_vals * f_hat u = self.ifft(u_hat) else: print("⚙️ 1D Kohn-Nirenberg Quantification") x, xi = symbols('x xi', real=True) R_func = lambdify((x, xi), R_symbol, 'numpy') # Still 2 args for uniformity u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func) elif self.dim == 2: if not R_symbol.has(x) and not R_symbol.has(y): print("⚡ Optimization: Symbol independent of x and y — direct product in 2D Fourier.") R_vals = np.vectorize(R_func)(self.KX, self.KY) u_hat = R_vals * f_hat u = self.ifft(u_hat) else: print("⚙️ 2D Kohn-Nirenberg Quantification") x, xi, y, eta = symbols('x xi y eta', real=True) R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy') # Still 2 args for uniformity u = self.kohn_nirenberg_fft(u_vals=rhs, symbol_func=R_func) self.u = u return u elif self.boundary_condition == 'dirichlet': if self.dim == 1: x, xi = symbols('x xi', real=True) R_func = lambdify((x, xi), R_symbol, 'numpy') u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=X, xi_grid=KX, symbol_func=R_func) elif self.dim == 2: x, xi, y, eta = symbols('x xi y eta', real=True) R_func = lambdify((x, y, xi, eta), R_symbol, 'numpy') u = self.kohn_nirenberg_nonperiodic(u_vals=rhs, x_grid=(self.x_grid, self.y_grid), xi_grid=(self.kx, self.ky), symbol_func=R_func) self.u = u return u else: raise ValueError( f"Invalid boundary condition '{self.boundary_condition}'. " "Supported types are 'periodic' and 'dirichlet'." ) def step_ETD_RK4(self, u)-
Perform one Exponential Time Differencing Runge-Kutta of 4th order (ETD-RK4) time step for first-order in time PDEs of the form:
∂ₜu = L u + N(u)where L is a linear operator (possibly nonlocal or pseudo-differential), and N is a nonlinear term treated via pseudo-spectral methods. This method evaluates the exponential integrator up to fourth-order accuracy in time.
The ETD-RK4 scheme uses four stages to approximate the integral of the variation-of-constants formula:
uⁿ⁺¹ = e^(L Δt) uⁿ + Δt ∫₀¹ e^(L Δt (1 - τ)) φ(N(u(τ))) dτwhere φ denotes the nonlinear contributions evaluated at intermediate stages.
Parameters u (np.ndarray): Current solution in real space (physical grid values).
Returns
np.ndarray- Updated solution in real space after one ETD-RK4 time step.
Notes: - The linear part L is diagonal in Fourier space and precomputed as self.L(k). - Nonlinear terms are evaluated in physical space and transformed via FFT. - The functions φ₁(z) and φ₂(z) are entire functions arising from the ETD scheme:
φ₁(z) = (eᶻ - 1)/z if z ≠ 0 = 1 if z = 0 φ₂(z) = (eᶻ - 1 - z)/z² if z ≠ 0 = ½ if z = 0- This implementation assumes periodic boundary conditions and uses spectral differentiation via FFT.
- See Hochbruck & Ostermann (2010) for theoretical background on exponential integrators.
See Also: step_ETD_RK4_order2 : For second-order in time equations. psiOp_apply : For applying pseudo-differential operators. apply_nonlinear : For handling nonlinear terms in the PDE.
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def step_ETD_RK4(self, u): """ Perform one Exponential Time Differencing Runge-Kutta of 4th order (ETD-RK4) time step for first-order in time PDEs of the form: ∂ₜu = L u + N(u) where L is a linear operator (possibly nonlocal or pseudo-differential), and N is a nonlinear term treated via pseudo-spectral methods. This method evaluates the exponential integrator up to fourth-order accuracy in time. The ETD-RK4 scheme uses four stages to approximate the integral of the variation-of-constants formula: uⁿ⁺¹ = e^(L Δt) uⁿ + Δt ∫₀¹ e^(L Δt (1 - τ)) φ(N(u(τ))) dτ where φ denotes the nonlinear contributions evaluated at intermediate stages. Parameters u (np.ndarray): Current solution in real space (physical grid values). Returns: np.ndarray: Updated solution in real space after one ETD-RK4 time step. Notes: - The linear part L is diagonal in Fourier space and precomputed as self.L(k). - Nonlinear terms are evaluated in physical space and transformed via FFT. - The functions φ₁(z) and φ₂(z) are entire functions arising from the ETD scheme: φ₁(z) = (eᶻ - 1)/z if z ≠ 0 = 1 if z = 0 φ₂(z) = (eᶻ - 1 - z)/z² if z ≠ 0 = ½ if z = 0 - This implementation assumes periodic boundary conditions and uses spectral differentiation via FFT. - See Hochbruck & Ostermann (2010) for theoretical background on exponential integrators. See Also: step_ETD_RK4_order2 : For second-order in time equations. psiOp_apply : For applying pseudo-differential operators. apply_nonlinear : For handling nonlinear terms in the PDE. """ dt = self.dt L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY) E = np.exp(dt * L_fft) E2 = np.exp(dt * L_fft / 2) def phi1(z): return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1) / z, 1.0) def phi2(z): return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1 - z) / z**2, 0.5) phi1_dtL = phi1(dt * L_fft) phi2_dtL = phi2(dt * L_fft) fft = self.fft ifft = self.ifft u_hat = fft(u) N1 = fft(self.apply_nonlinear(u)) a = ifft(E2 * (u_hat + 0.5 * dt * N1 * phi1_dtL)) N2 = fft(self.apply_nonlinear(a)) b = ifft(E2 * (u_hat + 0.5 * dt * N2 * phi1_dtL)) N3 = fft(self.apply_nonlinear(b)) c = ifft(E * (u_hat + dt * N3 * phi1_dtL)) N4 = fft(self.apply_nonlinear(c)) u_new_hat = E * u_hat + dt * ( N1 * phi1_dtL + 2 * (N2 + N3) * phi2_dtL + N4 * phi1_dtL ) / 6 return ifft(u_new_hat) def step_ETD_RK4_order2(self, u, v)-
Perform one time step of the Exponential Time Differencing Runge-Kutta 4th-order (ETD-RK4) scheme for second-order PDEs.
This method evolves the solution u and its time derivative v forward in time by one step using the ETD-RK4 integrator. It is designed for systems of the form:
∂ₜ²u = L u + N(u)where L is a linear operator and N is a nonlinear term computed via self.apply_nonlinear.
The exponential integrator handles the linear part exactly in Fourier space, while the nonlinear terms are integrated using a fourth-order Runge-Kutta-like approach. This ensures high accuracy and stability for stiff systems.
Parameters
u (np.ndarray): Current solution array in real space. v (np.ndarray): Current time derivative of the solution (∂ₜu) in real space.
Returns
tuple- (u_new, v_new), updated solution and its time derivative after one time step.
Notes
- Assumes periodic boundary conditions and uses FFT-based spectral methods.
- Handles both 1D and 2D problems seamlessly.
- Uses phi functions to compute exponential integrators efficiently.
- Suitable for wave equations and other second-order evolution equations with stiffness.
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def step_ETD_RK4_order2(self, u, v): """ Perform one time step of the Exponential Time Differencing Runge-Kutta 4th-order (ETD-RK4) scheme for second-order PDEs. This method evolves the solution u and its time derivative v forward in time by one step using the ETD-RK4 integrator. It is designed for systems of the form: ∂ₜ²u = L u + N(u) where L is a linear operator and N is a nonlinear term computed via self.apply_nonlinear. The exponential integrator handles the linear part exactly in Fourier space, while the nonlinear terms are integrated using a fourth-order Runge-Kutta-like approach. This ensures high accuracy and stability for stiff systems. Parameters: u (np.ndarray): Current solution array in real space. v (np.ndarray): Current time derivative of the solution (∂ₜu) in real space. Returns: tuple: (u_new, v_new), updated solution and its time derivative after one time step. Notes: - Assumes periodic boundary conditions and uses FFT-based spectral methods. - Handles both 1D and 2D problems seamlessly. - Uses phi functions to compute exponential integrators efficiently. - Suitable for wave equations and other second-order evolution equations with stiffness. """ dt = self.dt L_fft = self.L(self.KX) if self.dim == 1 else self.L(self.KX, self.KY) fft = self.fft ifft = self.ifft def rhs(u_val): return ifft(L_fft * fft(u_val)) + self.apply_nonlinear(u_val, is_v=False) # Stage A A = rhs(u) ua = u + 0.5 * dt * v va = v + 0.5 * dt * A # Stage B B = rhs(ua) ub = u + 0.5 * dt * va vb = v + 0.5 * dt * B # Stage C C = rhs(ub) uc = u + dt * vb # Stage D D = rhs(uc) # Final update u_new = u + dt * v + (dt**2 / 6.0) * (A + 2*B + 2*C + D) v_new = v + (dt / 6.0) * (A + 2*B + 2*C + D) return u_new, v_new def step_order1_with_psi(self, source_contribution)-
Perform one time step of a first-order evolution using a pseudo-differential operator.
This method updates the solution field using an exponential integrator or explicit Euler scheme, depending on boundary conditions and the structure of the pseudo-differential symbol. It supports: - Linear dynamics via pseudo-differential operator L (possibly nonlocal) - Nonlinear terms computed via spectral differentiation - External source contributions
The update follows three distinct computational paths:
-
Periodic boundaries + diagonalizable symbol
Symbol is constant in space → use direct Fourier-based exponential integrator:
uₙ₊₁ = e⁻ᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(−LΔt) ⋅ (N(uₙ) + F) -
Non-diagonalizable but spatially uniform symbol
General exponential time differencing of order 1:
uₙ₊₁ = eᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(LΔt) ⋅ (N(uₙ) + F) -
Spatially varying symbol
No frequency diagonalization available → use explicit Euler:
uₙ₊₁ = uₙ + Δt ⋅ (L(uₙ) + N(uₙ) + F)
where: L(uₙ) = linear part via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term Δt = time step size φ₁(z) = (eᶻ − 1)/z (with safe handling near z=0)
Boundary conditions are applied after each update to ensure consistency.
Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev.
Returns
np.ndarray- Updated solution array after one time step.
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def step_order1_with_psi(self, source_contribution): """ Perform one time step of a first-order evolution using a pseudo-differential operator. This method updates the solution field using an exponential integrator or explicit Euler scheme, depending on boundary conditions and the structure of the pseudo-differential symbol. It supports: - Linear dynamics via pseudo-differential operator L (possibly nonlocal) - Nonlinear terms computed via spectral differentiation - External source contributions The update follows **three distinct computational paths**: 1. **Periodic boundaries + diagonalizable symbol** Symbol is constant in space → use direct Fourier-based exponential integrator: uₙ₊₁ = e⁻ᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(−LΔt) ⋅ (N(uₙ) + F) 2. **Non-diagonalizable but spatially uniform symbol** General exponential time differencing of order 1: uₙ₊₁ = eᴸΔᵗ ⋅ uₙ + Δt ⋅ φ₁(LΔt) ⋅ (N(uₙ) + F) 3. **Spatially varying symbol** No frequency diagonalization available → use explicit Euler: uₙ₊₁ = uₙ + Δt ⋅ (L(uₙ) + N(uₙ) + F) where: L(uₙ) = linear part via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term Δt = time step size φ₁(z) = (eᶻ − 1)/z (with safe handling near z=0) Boundary conditions are applied after each update to ensure consistency. Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev. Returns: np.ndarray: Updated solution array after one time step. """ # Handling null source if np.isscalar(source_contribution): source = np.zeros_like(self.u_prev) else: source = source_contribution def spectral_filter(u, cutoff=0.8): if u.ndim == 1: u_hat = self.fft(u) N = len(u) k = fftfreq(N) mask = np.exp(-(k / cutoff)**8) return self.ifft(u_hat * mask).real elif u.ndim == 2: u_hat = self.fft(u) Ny, Nx = u.shape ky = fftfreq(Ny)[:, None] kx = fftfreq(Nx)[None, :] k_squared = kx**2 + ky**2 mask = np.exp(-(np.sqrt(k_squared) / cutoff)**8) return self.ifft(u_hat * mask).real else: raise ValueError("Only 1D and 2D arrays are supported.") # Recalculate symbol if necessary if self.is_spatial: self.prepare_symbol_tables() # Recalculates self.combined_symbol # Case with FFT (symbol diagonalizable in Fourier space) if self.boundary_condition == 'periodic' and not self.is_spatial: u_hat = self.fft(self.u_prev) u_hat *= np.exp(-self.dt * self.combined_symbol) u_hat *= self.dealiasing_mask u_symb = self.ifft(u_hat) u_nl = self.apply_nonlinear(self.u_prev) u_new = u_symb + u_nl + source else: if not self.is_spatial: # General case with ETD1 u_nl = self.apply_nonlinear(self.u_prev) # Calculation of exp(dt * L) and phi1(dt * L) L_vals = self.combined_symbol # Uses the updated symbol exp_L = np.exp(-self.dt * L_vals) phi1_L = (exp_L - 1.0) / (self.dt * L_vals) phi1_L[np.isnan(phi1_L)] = 1.0 # Handling division by zero # Fourier transform u_hat = self.fft(self.u_prev) u_nl_hat = self.fft(u_nl) source_hat = self.fft(source) # Assembling the solution in Fourier space u_hat_new = exp_L * u_hat + self.dt * phi1_L * (u_nl_hat + source_hat) u_new = self.ifft(u_hat_new) else: # if the symbol depends on spatial variables : Euler method Lu_prev = self.apply_psiOp(self.u_prev) u_nl = self.apply_nonlinear(self.u_prev) u_new = self.u_prev + self.dt * (Lu_prev + u_nl + source) u_new = spectral_filter(u_new, cutoff=self.dealiasing_ratio) # Applying boundary conditions self.apply_boundary(u_new) return u_new -
def step_order2_with_psi(self, source_contribution)-
Perform one time step of a second-order time evolution using a pseudo-differential operator.
This method updates the solution field using a second-order accurate scheme suitable for wave-like equations. The update includes contributions from: - Linear dynamics via a pseudo-differential operator (e.g., dispersion or stiffness) - Nonlinear terms computed via spectral differentiation - External source contributions
Discretization follows a leapfrog-style finite difference in time:
uₙ₊₁ = 2uₙ − uₙ₋₁ + Δt² ⋅ (L(uₙ) + N(uₙ) + F)where: L(uₙ) = linear part evaluated via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term at current time step Δt = time step size
Boundary conditions are applied after each update to ensure consistency.
Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev.
Returns
np.ndarray- Updated solution array after one time step.
Expand source code
def step_order2_with_psi(self, source_contribution): """ Perform one time step of a second-order time evolution using a pseudo-differential operator. This method updates the solution field using a second-order accurate scheme suitable for wave-like equations. The update includes contributions from: - Linear dynamics via a pseudo-differential operator (e.g., dispersion or stiffness) - Nonlinear terms computed via spectral differentiation - External source contributions Discretization follows a leapfrog-style finite difference in time: uₙ₊₁ = 2uₙ − uₙ₋₁ + Δt² ⋅ (L(uₙ) + N(uₙ) + F) where: L(uₙ) = linear part evaluated via pseudo-differential operator N(uₙ) = nonlinear contribution at current time step F = external source term at current time step Δt = time step size Boundary conditions are applied after each update to ensure consistency. Parameters source_contribution (np.ndarray): Array representing the external source term at current time step. Must match the spatial dimensions of self.u_prev. Returns: np.ndarray: Updated solution array after one time step. """ Lu_prev = self.apply_psiOp(self.u_prev) rhs_nl = self.apply_nonlinear(self.u_prev, is_v=False) u_new = 2 * self.u_prev - self.u_prev2 + self.dt ** 2 * (Lu_prev + rhs_nl + source_contribution) self.apply_boundary(u_new) self.u_prev2 = self.u_prev self.u_prev = u_new self.u = u_new return u_new def test(self, u_exact, t_eval=None, norm='relative', threshold=0.01, plot=True, component='real')-
Test the solver against an exact solution.
This method quantitatively compares the numerical solution with a provided exact solution at a specified time using either relative or absolute error norms. It supports both stationary and time-dependent problems in 1D and 2D. If enabled, it also generates plots of the solution, exact solution, and pointwise error.
Parameters
u_exact:callable- Exact solution function taking spatial coordinates and optionally time as arguments.
t_eval:float, optional- Time at which to compare solutions. For non-stationary problems, defaults to final time Lt. Ignored for stationary problems.
norm:str {'relative', 'absolute'}- Type of error norm used in comparison.
threshold:float- Acceptable error threshold; raises an assertion if exceeded.
plot:bool- Whether to display visual comparison plots (default: True).
component:str {'real', 'imag', 'abs'}- Component of the solution to compare and visualize.
Raises
ValueError- If unsupported dimension is encountered or requested evaluation time exceeds simulation duration.
AssertionError- If computed error exceeds the given threshold.
Prints
- Information about the closest available frame to the requested evaluation time.
- Computed error value and comparison to threshold.
Notes
- For time-dependent problems, the solution is extracted from precomputed frames.
- Plots are adapted to spatial dimension: line plots for 1D, image plots for 2D.
- The method ensures consistent handling of real, imaginary, and magnitude components.
Expand source code
def test(self, u_exact, t_eval=None, norm='relative', threshold=1e-2, plot=True, component='real'): """ Test the solver against an exact solution. This method quantitatively compares the numerical solution with a provided exact solution at a specified time using either relative or absolute error norms. It supports both stationary and time-dependent problems in 1D and 2D. If enabled, it also generates plots of the solution, exact solution, and pointwise error. Parameters ---------- u_exact : callable Exact solution function taking spatial coordinates and optionally time as arguments. t_eval : float, optional Time at which to compare solutions. For non-stationary problems, defaults to final time Lt. Ignored for stationary problems. norm : str {'relative', 'absolute'} Type of error norm used in comparison. threshold : float Acceptable error threshold; raises an assertion if exceeded. plot : bool Whether to display visual comparison plots (default: True). component : str {'real', 'imag', 'abs'} Component of the solution to compare and visualize. Raises ------ ValueError If unsupported dimension is encountered or requested evaluation time exceeds simulation duration. AssertionError If computed error exceeds the given threshold. Prints ------ - Information about the closest available frame to the requested evaluation time. - Computed error value and comparison to threshold. Notes ----- - For time-dependent problems, the solution is extracted from precomputed frames. - Plots are adapted to spatial dimension: line plots for 1D, image plots for 2D. - The method ensures consistent handling of real, imaginary, and magnitude components. """ if self.is_stationary: print("Testing a stationary solution.") u_num = self.u # Compute exact solution if self.dim == 1: u_ex = u_exact(self.X) elif self.dim == 2: u_ex = u_exact(self.X, self.Y) else: raise ValueError("Unsupported dimension.") actual_t = None else: if t_eval is None: t_eval = self.Lt save_interval = max(1, self.Nt // self.n_frames) frame_times = np.arange(0, self.Lt + self.dt, save_interval * self.dt) frame_index = np.argmin(np.abs(frame_times - t_eval)) actual_t = frame_times[frame_index] print(f"Closest available time to t_eval={t_eval}: {actual_t}") if frame_index >= len(self.frames): raise ValueError(f"Time t = {t_eval} exceeds simulation duration.") u_num = self.frames[frame_index] # Compute exact solution at the actual time if self.dim == 1: u_ex = u_exact(self.X, actual_t) elif self.dim == 2: u_ex = u_exact(self.X, self.Y, actual_t) else: raise ValueError("Unsupported dimension.") # Select component if component == 'real': diff = np.real(u_num) - np.real(u_ex) ref = np.real(u_ex) elif component == 'imag': diff = np.imag(u_num) - np.imag(u_ex) ref = np.imag(u_ex) elif component == 'abs': diff = np.abs(u_num) - np.abs(u_ex) ref = np.abs(u_ex) else: raise ValueError("Invalid component.") # Compute error if norm == 'relative': error = np.linalg.norm(diff) / np.linalg.norm(ref) elif norm == 'absolute': error = np.linalg.norm(diff) else: raise ValueError("Unknown norm type.") label_time = f"t = {actual_t}" if actual_t is not None else "" print(f"Test error {label_time}: {error:.3e}") assert error < threshold, f"Error too large {label_time}: {error:.3e}" # Plot if plot: if self.dim == 1: plt.figure(figsize=(12, 6)) plt.subplot(2, 1, 1) plt.plot(self.X, np.real(u_num), label='Numerical') plt.plot(self.X, np.real(u_ex), '--', label='Exact') plt.title(f'Solution {label_time}, error = {error:.2e}') plt.legend() plt.grid() plt.subplot(2, 1, 2) plt.plot(self.X, np.abs(diff), color='red') plt.title('Absolute Error') plt.grid() plt.tight_layout() plt.show() else: plt.figure(figsize=(15, 5)) plt.subplot(1, 3, 1) plt.title("Numerical Solution") plt.imshow(np.abs(u_num), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis') plt.colorbar() plt.subplot(1, 3, 2) plt.title("Exact Solution") plt.imshow(np.abs(u_ex), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='viridis') plt.colorbar() plt.subplot(1, 3, 3) plt.title(f"Error (Norm = {error:.2e})") plt.imshow(np.abs(diff), origin='lower', extent=[0, self.Lx, 0, self.Ly], cmap='inferno') plt.colorbar() plt.tight_layout() plt.show() def total_symbol_expr(self)-
Compute the total pseudo-differential symbol expression from all pseudo_terms.
This method constructs the full symbol of the pseudo-differential operator by summing up all coefficient-weighted symbolic expressions.
The result is cached in self.symbol_expr to avoid recomputation.
Returns
sympy.Expr- The combined symbol expression, representing the full pseudo-differential operator in symbolic form.
Example
Given pseudo_terms = [(2, ξ²), (1, x·ξ)], this returns 2·ξ² + x·ξ.
Expand source code
def total_symbol_expr(self): """ Compute the total pseudo-differential symbol expression from all pseudo_terms. This method constructs the full symbol of the pseudo-differential operator by summing up all coefficient-weighted symbolic expressions. The result is cached in self.symbol_expr to avoid recomputation. Returns: sympy.Expr: The combined symbol expression, representing the full pseudo-differential operator in symbolic form. Example: Given pseudo_terms = [(2, ξ²), (1, x·ξ)], this returns 2·ξ² + x·ξ. """ if not hasattr(self, '_symbol_expr'): self.symbol_expr = sum(coeff * expr for coeff, expr in self.pseudo_terms) return self.symbol_expr
class PseudoDifferentialOperator (expr, vars_x, var_u=None, mode='symbol')-
Pseudo-differential operator with dynamic symbol evaluation on spatial grids. Supports both 1D and 2D operators, and can be defined explicitly (symbol mode) or extracted automatically from symbolic equations (auto mode).
Parameters
expr:sympy expression- Symbolic expression representing the pseudo-differential symbol.
vars_x:listofsympy symbols- Spatial variables (e.g., [x] for 1D, [x, y] for 2D).
var_u:sympy function, optional- Function u(x, t) used in auto mode to extract the operator symbol.
mode:str, {'symbol', 'auto'}-
- 'symbol': directly uses expr as the operator symbol.
- 'auto': computes the symbol automatically by applying expr to exp(i x ξ).
Attributes
dim:int- Spatial dimension (1 or 2).
fft,ifft:callable- Fast Fourier transform and inverse (scipy.fft or scipy.fft2).
p_func:callable- Evaluated symbol function ready for numerical use.
Notes
- In 'symbol' mode,
exprshould be expressed in terms of spatial variables and frequency variables (ξ, η). - In 'auto' mode, the symbol is derived by applying the differential expression to a complex exponential.
- Frequency variables are internally named 'xi' and 'eta' for consistency.
- Uses numpy for numerical evaluation and scipy.fft for FFT operations.
Examples
>>> # Example 1: 1D Laplacian operator (symbol mode) >>> from sympy import symbols >>> x, xi = symbols('x xi', real=True) >>> op = PseudoDifferentialOperator(expr=xi**2, vars_x=[x], mode='symbol')>>> # Example 2: 1D transport operator (auto mode) >>> from sympy import Function >>> u = Function('u') >>> expr = u(x).diff(x) >>> op = PseudoDifferentialOperator(expr=expr, vars_x=[x], var_u=u(x), mode='auto')Expand source code
class PseudoDifferentialOperator: """ Pseudo-differential operator with dynamic symbol evaluation on spatial grids. Supports both 1D and 2D operators, and can be defined explicitly (symbol mode) or extracted automatically from symbolic equations (auto mode). Parameters ---------- expr : sympy expression Symbolic expression representing the pseudo-differential symbol. vars_x : list of sympy symbols Spatial variables (e.g., [x] for 1D, [x, y] for 2D). var_u : sympy function, optional Function u(x, t) used in auto mode to extract the operator symbol. mode : str, {'symbol', 'auto'} - 'symbol': directly uses expr as the operator symbol. - 'auto': computes the symbol automatically by applying expr to exp(i x ξ). Attributes ---------- dim : int Spatial dimension (1 or 2). fft, ifft : callable Fast Fourier transform and inverse (scipy.fft or scipy.fft2). p_func : callable Evaluated symbol function ready for numerical use. Notes ----- - In 'symbol' mode, `expr` should be expressed in terms of spatial variables and frequency variables (ξ, η). - In 'auto' mode, the symbol is derived by applying the differential expression to a complex exponential. - Frequency variables are internally named 'xi' and 'eta' for consistency. - Uses numpy for numerical evaluation and scipy.fft for FFT operations. Examples -------- >>> # Example 1: 1D Laplacian operator (symbol mode) >>> from sympy import symbols >>> x, xi = symbols('x xi', real=True) >>> op = PseudoDifferentialOperator(expr=xi**2, vars_x=[x], mode='symbol') >>> # Example 2: 1D transport operator (auto mode) >>> from sympy import Function >>> u = Function('u') >>> expr = u(x).diff(x) >>> op = PseudoDifferentialOperator(expr=expr, vars_x=[x], var_u=u(x), mode='auto') """ def __init__(self, expr, vars_x, var_u=None, mode='symbol'): self.dim = len(vars_x) self.mode = mode self.symbol_cached = None self.expr = expr self.vars_x = vars_x if self.dim == 1: x, = vars_x xi_internal = symbols('xi', real=True) expr = expr.subs(symbols('xi', real=True), xi_internal) self.fft = partial(fft, workers=FFT_WORKERS) self.ifft = partial(ifft, workers=FFT_WORKERS) if mode == 'symbol': self.p_func = lambdify((x, xi_internal), expr, 'numpy') self.symbol = expr elif mode == 'auto': if var_u is None: raise ValueError("var_u must be provided in mode='auto'") exp_i = exp(I * x * xi_internal) P_ei = expr.subs(var_u, exp_i) symbol = simplify(P_ei / exp_i) symbol = expand(symbol) self.symbol = symbol self.p_func = lambdify((x, xi_internal), symbol, 'numpy') else: raise ValueError("mode must be 'auto' or 'symbol'") elif self.dim == 2: x, y = vars_x xi_internal, eta_internal = symbols('xi eta', real=True) expr = expr.subs(symbols('xi', real=True), xi_internal) expr = expr.subs(symbols('eta', real=True), eta_internal) self.fft = partial(fft2, workers=FFT_WORKERS) self.ifft = partial(ifft2, workers=FFT_WORKERS) if mode == 'symbol': self.symbol = expr self.p_func = lambdify((x, y, xi_internal, eta_internal), expr, 'numpy') elif mode == 'auto': if var_u is None: raise ValueError("var_u must be provided in mode='auto'") exp_i = exp(I * (x * xi_internal + y * eta_internal)) P_ei = expr.subs(var_u, exp_i) symbol = simplify(P_ei / exp_i) symbol = expand(symbol) self.symbol = symbol self.p_func = lambdify((x, y, xi_internal, eta_internal), symbol, 'numpy') else: raise ValueError("mode must be 'auto' or 'symbol'") else: raise NotImplementedError("Only 1D and 2D supported") print("\nsymbol = ") pprint(expr, num_columns=150) def evaluate(self, X, Y, KX, KY, cache=True): """ Evaluate the pseudo-differential operator's symbol on a grid of spatial and frequency coordinates. The method dynamically selects between 1D and 2D evaluation based on the spatial dimension. If caching is enabled and a cached symbol exists, it returns the cached result to avoid recomputation. Parameters ---------- X, Y : ndarray Spatial grid coordinates. In 1D, Y is ignored. KX, KY : ndarray Frequency grid coordinates. In 1D, KY is ignored. cache : bool, default=True If True, stores the computed symbol for reuse in subsequent calls to avoid redundant computation. Returns ------- ndarray Evaluated symbol values over the input grid. Shape matches the input spatial/frequency grids. Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. """ if cache and self.symbol_cached is not None: return self.symbol_cached if self.dim == 1: symbol = self.p_func(X, KX) elif self.dim == 2: symbol = self.p_func(X, Y, KX, KY) if cache: self.symbol_cached = symbol return symbol def clear_cache(self): """ Clear cached symbol evaluations. """ self.symbol_cached = None def principal_symbol(self, order=1): """ Compute the leading homogeneous component of the pseudo-differential symbol. This method extracts the principal part of the symbol, which is the dominant term under high-frequency asymptotics (|ξ| → ∞). The expansion is performed in polar coordinates for 2D symbols to maintain rotational symmetry, then converted back to Cartesian form. Parameters ---------- order : int Order of the asymptotic expansion in powers of 1/ρ, where ρ = |ξ| in 1D or ρ = sqrt(ξ² + η²) in 2D. Only the leading-order term is returned. Returns ------- sympy.Expr The principal symbol component, homogeneous of degree `m - order`, where `m` is the original symbol's order. Notes: - In 1D, uses direct series expansion in ξ. - In 2D, expands in radial variable ρ while preserving angular dependence. - Useful for microlocal analysis and constructing parametrices. """ p = self.expr if self.dim == 1: xi = symbols('xi', real=True, positive=True) return simplify(series(p, xi, oo, n=order).removeO()) elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) # Homogeneous radial expansion: we set (ξ, η) = ρ (cosθ, sinθ) rho, theta = symbols('rho theta', real=True, positive=True) p_rho = p.subs({xi: rho * cos(theta), eta: rho * sin(theta)}) expansion = series(p_rho, rho, oo, n=order).removeO() # Revert back to (ξ, η) expansion_cart = expansion.subs({rho: sqrt(xi**2 + eta**2), cos(theta): xi / sqrt(xi**2 + eta**2), sin(theta): eta / sqrt(xi**2 + eta**2)}) return simplify(powdenest(expansion_cart, force=True)) def is_homogeneous(self, tol=1e-10): """ Check whether the symbol is homogeneous in the frequency variables. Returns ------- (bool, Rational or float or None) Tuple (is_homogeneous, degree) where: - is_homogeneous: True if the symbol satisfies p(λξ, λη) = λ^m * p(ξ, η) - degree: the detected degree m if homogeneous, or None """ from sympy import symbols, simplify, expand, Eq from sympy.abc import l if self.dim == 1: xi = symbols('xi', real=True, positive=True) l = symbols('l', real=True, positive=True) p = self.expr p_scaled = p.subs(xi, l * xi) ratio = simplify(p_scaled / p) if ratio.has(xi): return False, None try: deg = simplify(ratio).as_base_exp()[1] return True, deg except Exception: return False, None elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) l = symbols('l', real=True, positive=True) p = self.expr p_scaled = p.subs({xi: l * xi, eta: l * eta}) ratio = simplify(p_scaled / p) # If ratio == l**m with no (xi, eta) left, it's homogeneous if ratio.has(xi, eta): return False, None try: base, exp = ratio.as_base_exp() if base == l: return True, exp except Exception: pass return False, None def symbol_order(self, max_order=10, tol=1e-3): """ Estimate the homogeneity order of the pseudo-differential symbol in high-frequency asymptotics. This method attempts to determine the leading-order behavior of the symbol p(x, ξ) or p(x, y, ξ, η) as |ξ| → ∞ (in 1D) or |(ξ, η)| → ∞ (in 2D). The returned value represents the asymptotic growth or decay rate, which is essential for understanding the regularity and mapping properties of the corresponding operator. The function uses symbolic preprocessing to ensure proper factorization of frequency variables, especially in sqrt and power expressions, to avoid erroneous order detection (e.g., due to hidden scaling). Parameters ---------- max_order : int, optional Maximum number of terms to consider in the series expansion. Default is 10. tol : float, optional Tolerance threshold for evaluating the coefficient magnitude. If the coefficient is too small, the detected order may be discarded. Default is 1e-3. Returns ------- float or None - If the symbol is homogeneous, returns its exact homogeneity degree as a float. - Otherwise, estimates the dominant asymptotic order from leading terms in the expansion. - Returns None if no valid order could be determined. Notes ----- - In 1D: Two strategies are used: 1. Expand directly in xi at infinity. 2. Substitute xi = 1/z and expand around z = 0. - In 2D: - Transform the symbol into polar coordinates: (xi, eta) = rho*(cos(theta), sin(theta)). - Expand in rho at infinity, then extract the leading term's power. - An alternative substitution using 1/z is also tried if the first method fails. - Preprocessing steps: - Sqrt expressions involving frequencies are rewritten to isolate the leading variable. - Power expressions are factored explicitly to ensure correct symbolic scaling. - If the symbol is not homogeneous, a warning is issued, and the result should be interpreted with care. - For non-homogeneous symbols, only the principal asymptotic term is considered. Raises ------ NotImplementedError If the spatial dimension is neither 1 nor 2. """ from sympy import ( symbols, series, simplify, sqrt, cos, sin, oo, powdenest, radsimp, expand, expand_power_base ) def preprocess_sqrt(expr, freq): return expr.replace( lambda e: e.func == sqrt and freq in e.free_symbols, lambda e: freq * sqrt(1 + (e.args[0] - freq**2) / freq**2) ) def preprocess_power(expr, freq): return expr.replace( lambda e: e.is_Pow and freq in e.free_symbols, lambda e: freq**e.exp * (1 + e.base / freq**e.base.as_powers_dict().get(freq, 0))**e.exp ) def validate_order(power, coeff, vars_x, tol): if power is None: return None if any(v in coeff.free_symbols for v in vars_x): print("⚠️ Coefficient depends on spatial variables; ignoring") return None try: coeff_val = abs(float(coeff.evalf())) if coeff_val < tol: print(f"⚠️ Coefficient too small ({coeff_val:.2e} < {tol})") return None except Exception as e: print(f"⚠️ Coefficient evaluation failed: {e}") return None return int(power) if power == int(power) else float(power) # Homogeneity check is_homog, degree = self.is_homogeneous() if is_homog: return float(degree) else: print("⚠️ The symbol is not homogeneous. The asymptotic order is not well defined.") if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True, positive=True) try: print("1D symbol_order - method 1") expr = preprocess_sqrt(self.expr, xi) s = series(expr, xi, oo, n=max_order).removeO() lead = simplify(powdenest(s.as_leading_term(xi), force=True)) power = lead.as_powers_dict().get(xi, None) coeff = lead / xi**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x], tol) if order is not None: return order except Exception: pass try: print("1D symbol_order - method 2") z = symbols('z', real=True, positive=True) expr_z = preprocess_sqrt(self.expr.subs(xi, 1/z), 1/z) s = series(expr_z, z, 0, n=max_order).removeO() lead = simplify(powdenest(s.as_leading_term(z), force=True)) power = lead.as_powers_dict().get(z, None) coeff = lead / z**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x], tol) if order is not None: return -order except Exception as e: print(f"⚠️ fallback z failed: {e}") return None elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True, positive=True) rho, theta = symbols('rho theta', real=True, positive=True) try: print("2D symbol_order - method 1") p_rho = self.expr.subs({xi: rho * cos(theta), eta: rho * sin(theta)}) p_rho = preprocess_power(preprocess_sqrt(p_rho, rho), rho) s = series(simplify(p_rho), rho, oo, n=max_order).removeO() lead = radsimp(simplify(powdenest(s.as_leading_term(rho), force=True))) power = lead.as_powers_dict().get(rho, None) coeff = lead / rho**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x, y], tol) if order is not None: return order except Exception as e: print(f"⚠️ polar expansion failed: {e}") try: print("2D symbol_order - method 2") z = symbols('z', real=True, positive=True) xi_eta = {xi: (1/z) * cos(theta), eta: (1/z) * sin(theta)} p_rho = preprocess_sqrt(self.expr.subs(xi_eta), 1/z) s = series(simplify(p_rho), z, 0, n=max_order).removeO() lead = radsimp(simplify(powdenest(s.as_leading_term(z), force=True))) power = lead.as_powers_dict().get(z, None) coeff = lead / z**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x, y], tol) if order is not None: return -order except Exception as e: print(f"⚠️ fallback z (2D) failed: {e}") return None else: raise NotImplementedError("Only 1D and 2D supported.") def asymptotic_expansion(self, order=3): """ Compute the asymptotic expansion of the symbol as |ξ| → ∞ (high-frequency regime). This method expands the pseudo-differential symbol in inverse powers of the frequency variable(s), either in 1D or 2D. It handles both polynomial and exponential symbols by performing a series expansion in 1/|ξ| up to the specified order. The expansion is performed directly in Cartesian coordinates for 1D symbols. For 2D symbols, the method uses polar coordinates (ρ, θ) to perform the expansion at infinity in ρ, then converts the result back to Cartesian coordinates. Parameters ---------- order : int, optional Maximum order of the asymptotic expansion. Default is 3. Returns ------- sympy.Expr The asymptotic expansion of the symbol up to the given order, expressed in Cartesian coordinates. If expansion fails, returns the original unexpanded symbol. Notes: - In 1D: expansion is performed directly in terms of ξ. - In 2D: the symbol is first rewritten in polar coordinates (ρ,θ), expanded asymptotically in ρ → ∞, then converted back to Cartesian coordinates (ξ,η). - Handles special case when the symbol is an exponential function by expanding its argument. - Symbolic normalization is applied early (via `simplify`) for 2D expressions to improve convergence. - Robust to failures: catches exceptions and issues warnings instead of raising errors. - Final expression is simplified using `powdenest` and `expand` for improved readability. """ p = self.expr if self.dim == 1: xi = symbols('xi', real=True, positive=True) try: # Case: exponential function if p.func == exp and len(p.args) == 1: arg = p.args[0] arg_series = series(arg, xi, oo, n=order).removeO() expanded = series(exp(expand(arg_series)), xi, oo, n=order).removeO() return simplify(powdenest(expanded, force=True)) else: expanded = series(p, xi, oo, n=order).removeO() return simplify(powdenest(expanded, force=True)) except Exception as e: print(f"Warning: 1D expansion failed: {e}") return p elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) rho, theta = symbols('rho theta', real=True, positive=True) # Normalize before substitution p = simplify(p) # Substitute polar coordinates p_polar = p.subs({ xi: rho * cos(theta), eta: rho * sin(theta) }) try: # Handle exponentials if p_polar.func == exp and len(p_polar.args) == 1: arg = p_polar.args[0] arg_series = series(arg, rho, oo, n=order).removeO() expanded = series(exp(expand(arg_series)), rho, oo, n=order).removeO() else: expanded = series(p_polar, rho, oo, n=order).removeO() # Convert back to Cartesian norm = sqrt(xi**2 + eta**2) expansion_cart = expanded.subs({ rho: norm, cos(theta): xi / norm, sin(theta): eta / norm }) # Final simplifications result = simplify(powdenest(expansion_cart, force=True)) result = expand(result) return result except Exception as e: print(f"Warning: 2D expansion failed: {e}") return p def compose_asymptotic(self, other, order=1): """ Compose this pseudo-differential operator with another using formal asymptotic expansion. This method computes the composition symbol via an asymptotic expansion in powers of derivatives, following the symbolic calculus of pseudo-differential operators. The composition is performed up to the specified order and respects the dimensionality (1D or 2D) of the operators. Parameters ---------- other : PseudoDifferentialOperator The pseudo-differential operator to compose with this one. order : int, default=1 Maximum order of the asymptotic expansion. Higher values include more terms in the symbolic composition, increasing accuracy at the cost of complexity. Returns ------- sympy.Expr Symbolic expression representing the asymptotic expansion of the composed operator. Notes ----- - In 1D, the composition uses the formula: (p ∘ q)(x, ξ) ~ Σₙ (1/n!) ∂_ξⁿ p(x, ξ) ∂_xⁿ q(x, ξ) (i)^{-n} - In 2D, the multi-index generalization is used: (p ∘ q)(x, y, ξ, η) ~ Σₙ Σᵢ (1/(i! j!)) ∂_ξⁱ∂_ηʲ p ∂_xⁱ∂_yʲ q (i)^{-n}, where n = i + j. - This expansion is valid for symbols admitting an asymptotic series representation. - Operators must be defined on the same spatial domain (same dimension). """ assert self.dim == other.dim, "Operator dimensions must match" p, q = self.expr, other.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) result = 0 for n in range(order + 1): term = (1 / factorial(n)) * diff(p, xi, n) * diff(q, x, n) * (1j)**(-n) result += term elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) result = 0 for n in range(order + 1): for i in range(n + 1): j = n - i term = (1 / (factorial(i) * factorial(j))) * \ diff(p, xi, i, eta, j) * diff(q, x, i, y, j) * (1j)**(-n) result += term return result def right_inverse_asymptotic(self, order=1): """ Construct a formal right inverse R of the pseudo-differential operator P such that the composition P ∘ R equals the identity plus a smoothing operator of order -order. This method computes an asymptotic expansion for the right inverse using recursive corrections based on derivatives of the symbol p(x, ξ) and lower-order terms of R. Parameters ---------- order : int Number of terms to include in the asymptotic expansion. Higher values improve approximation at the cost of complexity and computational effort. Returns ------- sympy.Expr The symbolic expression representing the formal right inverse R(x, ξ), which satisfies: P ∘ R = Id + O(⟨ξ⟩^{-order}), where ⟨ξ⟩ = (1 + |ξ|²)^{1/2}. Notes ----- - In 1D: The recursion involves spatial derivatives of R and derivatives of p with respect to ξ. - In 2D: The multi-index generalization is used with mixed derivatives in ξ and η. - The construction relies on the non-vanishing of the principal symbol p to ensure invertibility. - Each term in the expansion corresponds to higher-order corrections involving commutators between the operator P and the current approximation of R. """ p = self.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) r = 1 / p.subs(xi, xi) # r0 R = r for n in range(1, order + 1): term = 0 for k in range(1, n + 1): coeff = (1j)**(-k) / factorial(k) inner = diff(p, xi, k) * diff(R, x, k) term += coeff * inner R = R - r * term elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) r = 1 / p.subs({xi: xi, eta: eta}) R = r for n in range(1, order + 1): term = 0 for k1 in range(n + 1): for k2 in range(n + 1 - k1): if k1 + k2 == 0: continue coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2)) dp = diff(p, xi, k1, eta, k2) dR = diff(R, x, k1, y, k2) term += coeff * dp * dR R = R - r * term return R def left_inverse_asymptotic(self, order=1): """ Construct a formal left inverse L such that the composition L ∘ P equals the identity operator up to terms of order ξ^{-order}. This expansion is performed asymptotically at infinity in the frequency variable(s). The left inverse is built iteratively using symbolic differentiation and the method of asymptotic expansions for pseudo-differential operators. It ensures that: L(P(x,ξ),x,D) ∘ P(x,D) = Id + smoothing operator of order -order Parameters ---------- order : int, optional Maximum number of terms in the asymptotic expansion (default is 1). Higher values yield more accurate inverses at the cost of increased computational complexity. Returns ------- sympy.Expr Symbolic expression representing the principal symbol of the formal left inverse operator L(x,ξ). This expression depends on spatial variables and frequencies, and includes correction terms up to the specified order. Notes ----- - In 1D: Uses recursive application of the Leibniz formula for symbols. - In 2D: Generalizes to multi-indices for mixed derivatives in (x,y) and (ξ,η). - Each term involves combinations of derivatives of the original symbol p(x,ξ) and previously computed terms of the inverse. - Coefficients include powers of 1j (i) and factorial normalization for derivative terms. """ p = self.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) l = 1 / p.subs(xi, xi) L = l for n in range(1, order + 1): term = 0 for k in range(1, n + 1): coeff = (1j)**(-k) / factorial(k) inner = diff(L, xi, k) * diff(p, x, k) term += coeff * inner L = L - term * l elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) l = 1 / p.subs({xi: xi, eta: eta}) L = l for n in range(1, order + 1): term = 0 for k1 in range(n + 1): for k2 in range(n + 1 - k1): if k1 + k2 == 0: continue coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2)) dp = diff(p, x, k1, y, k2) dL = diff(L, xi, k1, eta, k2) term += coeff * dL * dp L = L - term * l return L def formal_adjoint(self): """ Compute the formal adjoint symbol P* of the pseudo-differential operator. The adjoint is defined such that for any test functions u and v, ⟨P u, v⟩ = ⟨u, P* v⟩ holds in the distributional sense. This is obtained by taking the complex conjugate of the symbol and expanding it asymptotically at infinity to ensure proper behavior under integration by parts. Returns ------- sympy.Expr The adjoint symbol P*(x, ξ) in 1D or P*(x, y, ξ, η) in 2D. Notes: - In 1D, the expansion is performed in powers of 1/|ξ|. - In 2D, the expansion is radial in |ξ| = sqrt(ξ² + η²). - This method ensures symbolic simplifications for readability and efficiency. """ p = self.expr if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) p_star = conjugate(p) p_star = simplify(series(p_star, xi, oo, n=6).removeO()) return p_star elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) p_star = conjugate(p) p_star = simplify(series(p_star, sqrt(xi**2 + eta**2), oo, n=6).removeO()) return p_star def symplectic_flow(self): """ Compute the Hamiltonian vector field associated with the principal symbol. This method derives the canonical equations of motion for the phase space variables (x, ξ) in 1D or (x, y, ξ, η) in 2D, based on the Hamiltonian formalism. These describe how position and frequency variables evolve under the flow generated by the symbol. Returns ------- dict A dictionary containing the components of the Hamiltonian vector field: - In 1D: keys are 'dx/dt' and 'dxi/dt', corresponding to dx/dt = ∂p/∂ξ and dξ/dt = -∂p/∂x. - In 2D: keys are 'dx/dt', 'dy/dt', 'dxi/dt', and 'deta/dt', with similar definitions: dx/dt = ∂p/∂ξ, dy/dt = ∂p/∂η, dξ/dt = -∂p/∂x, dη/dt = -∂p/∂y. Notes ----- - The Hamiltonian here is the principal symbol p(x, ξ) itself. - This flow preserves the symplectic structure of phase space. """ if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) print("x = ", x) print("xi = ", xi) return { 'dx/dt': diff(self.symbol, xi), 'dxi/dt': -diff(self.symbol, x) } elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) return { 'dx/dt': diff(self.symbol, xi), 'dy/dt': diff(self.symbol, eta), 'dxi/dt': -diff(self.symbol, x), 'deta/dt': -diff(self.symbol, y) } def is_elliptic_numerically(self, x_grid, xi_grid, threshold=1e-8): """ Check if the pseudo-differential symbol p(x, ξ) is elliptic over a given grid. A symbol is considered elliptic if its magnitude |p(x, ξ)| remains bounded away from zero across all points in the spatial-frequency domain. This method evaluates the symbol on a grid of spatial and frequency coordinates and checks whether its minimum absolute value exceeds a specified threshold. Resampling is applied to large grids to prevent excessive memory usage, particularly in 2D. Parameters ---------- x_grid : ndarray Spatial grid: either a 1D array (x) or a tuple of two 1D arrays (x, y). xi_grid : ndarray Frequency grid: either a 1D array (ξ) or a tuple of two 1D arrays (ξ, η). threshold : float, optional Minimum acceptable value for |p(x, ξ)|. If the smallest evaluated symbol value falls below this, the symbol is not considered elliptic. Returns ------- bool True if the symbol is elliptic on the resampled grid, False otherwise. """ RESAMPLE_SIZE = 32 # Reduced size to prevent memory explosion if self.dim == 1: x_vals = x_grid xi_vals = xi_grid # Resampling if necessary if len(x_vals) > RESAMPLE_SIZE: x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE) if len(xi_vals) > RESAMPLE_SIZE: xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') symbol_vals = self.p_func(X, XI) elif self.dim == 2: x_vals, y_vals = x_grid xi_vals, eta_vals = xi_grid # Spatial resampling if len(x_vals) > RESAMPLE_SIZE: x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE) if len(y_vals) > RESAMPLE_SIZE: y_vals = np.linspace(y_vals.min(), y_vals.max(), RESAMPLE_SIZE) # Frequency resampling if len(xi_vals) > RESAMPLE_SIZE: xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE) if len(eta_vals) > RESAMPLE_SIZE: eta_vals = np.linspace(eta_vals.min(), eta_vals.max(), RESAMPLE_SIZE) X, Y, XI, ETA = np.meshgrid(x_vals, y_vals, xi_vals, eta_vals, indexing='ij') symbol_vals = self.p_func(X, Y, XI, ETA) min_abs_val = np.min(np.abs(symbol_vals)) return min_abs_val > threshold def is_self_adjoint(self, tol=1e-10): """ Check whether the pseudo-differential operator is formally self-adjoint (Hermitian). A self-adjoint operator satisfies P = P*, where P* is the formal adjoint of P. This property is essential for ensuring real-valued eigenvalues and stable evolution in quantum mechanics and symmetric wave propagation. Parameters ---------- tol : float Tolerance for symbolic comparison between P and P*. Small numerical differences below this threshold are considered equal. Returns ------- bool True if the symbol p(x, ξ) equals its formal adjoint p*(x, ξ) within the given tolerance, indicating that the operator is self-adjoint. Notes: - The formal adjoint is computed via conjugation and asymptotic expansion at infinity in ξ. - Symbolic simplification is used to verify equality, ensuring robustness against superficial expression differences. """ p = self.expr p_star = self.formal_adjoint() return simplify(p - p_star).equals(0) def visualize_wavefront(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Visualize the wavefront set of the pseudo-differential operator's symbol by plotting |p(x, ξ)| or |p(x, y, ξ₀, η₀)|. The wavefront set captures the location and direction of singularities in the symbol p, which is crucial in microlocal analysis for understanding propagation of singularities and wave behavior. In 1D, this method displays |p(x, ξ)| over the phase space (x, ξ). In 2D, it fixes the frequency values (ξ₀, η₀) and visualizes |p(x, y, ξ₀, η₀)| over the spatial domain to highlight where the symbol interacts with the fixed frequency. Parameters ---------- x_grid : ndarray 1D array of spatial coordinates (x) used for evaluation. xi_grid : ndarray 1D array of frequency coordinates (ξ) used for visualization in 1D or slicing in 2D. y_grid : ndarray, optional 1D array of second spatial coordinate (y), used only in 2D. eta_grid : ndarray, optional 1D array of second frequency coordinate (η), not directly used but kept for API consistency. xi0 : float, default=0.0 Fixed value of ξ for slicing in 2D visualization. eta0 : float, default=0.0 Fixed value of η for slicing in 2D visualization. Notes ----- - In 1D: Displays a 2D color map of |p(x, ξ)| over the phase space with axes (x, ξ). - In 2D: Evaluates and plots |p(x, y, ξ₀, η₀)| at fixed frequencies (ξ₀, η₀) over the spatial grid (x, y). - Uses `imshow` for fast and scalable visualization with automatic aspect ratio adjustment. - A colormap ('viridis') is used to enhance contrast and readability of the magnitude variations. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid) symbol_vals = self.p_func(X, XI) plt.imshow(np.abs(symbol_vals), extent=[xi_grid.min(), xi_grid.max(), x_grid.min(), x_grid.max()], aspect='auto', origin='lower', cmap='viridis') elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid) XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.imshow(np.abs(symbol_vals), extent=[x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()], aspect='auto', origin='lower', cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('ξ/Position') plt.ylabel('η/Position') plt.title('Wavefront Set') plt.show() def visualize_fiber(self, x_grid, xi_grid, y0=0.0, x0=0.0): """ Plot the cotangent fiber structure at a fixed spatial point (x₀[, y₀]). This visualization shows how the symbol p(x, ξ) behaves on the cotangent fiber above a fixed spatial point. In microlocal analysis, this provides insight into the frequency content of the operator at that location. Parameters ---------- x_grid : ndarray Spatial grid values (1D) for evaluation in 1D case. xi_grid : ndarray Frequency grid values (1D) for evaluation in both 1D and 2D cases. x0 : float, optional Fixed x-coordinate of the base point in space (1D or 2D). y0 : float, optional Fixed y-coordinate of the base point in space (2D only). Notes ----- - In 1D: Displays |p(x, ξ)| over the (x, ξ) phase plane near the fixed point. - In 2D: Fixes (x₀, y₀) and evaluates p(x₀, y₀, ξ, η), showing the fiber over that point. - The color map represents the magnitude of the symbol, highlighting regions where it vanishes or becomes singular. Raises ------ NotImplementedError If called in 2D with missing or improperly formatted grids. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.contourf(X, XI, np.abs(symbol_vals), levels=50, cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('x (position)') plt.ylabel('ξ (frequency)') plt.title('Cotangent Fiber Structure') plt.show() elif self.dim == 2: xi_grid2, eta_grid2 = np.meshgrid(xi_grid, xi_grid) symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) plt.contourf(xi_grid, xi_grid, np.abs(symbol_vals), levels=50, cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('ξ') plt.ylabel('η') plt.title(f'Cotangent Fiber at x={x0}, y={y0}') plt.show() def visualize_symbol_amplitude(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Display the modulus |p(x, ξ)| or |p(x, y, ξ₀, η₀)| as a color map. This method visualizes the amplitude of the pseudodifferential operator's symbol in either 1D or 2D spatial configuration. In 2D, the frequency variables are fixed to specified values (ξ₀, η₀) for visualization purposes. Parameters ---------- x_grid, y_grid : ndarray Spatial grids over which to evaluate the symbol. y_grid is optional and used only in 2D. xi_grid, eta_grid : ndarray Frequency grids. In 2D, these define the domain over which the symbol is evaluated, but the visualization fixes ξ = ξ₀ and η = η₀. xi0, eta0 : float, optional Fixed frequency values for slicing in 2D visualization. Defaults to zero. Notes ----- - In 1D: Visualizes |p(x, ξ)| over the (x, ξ) grid. - In 2D: Visualizes |p(x, y, ξ₀, η₀)| at fixed frequencies ξ₀ and η₀. - The color intensity represents the magnitude of the symbol, highlighting regions where the symbol is large or small. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.pcolormesh(X, XI, np.abs(symbol_vals), shading='auto') plt.colorbar(label='|Symbol|') plt.xlabel('x') plt.ylabel('ξ') plt.title('Symbol Amplitude |p(x, ξ)|') plt.show() elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.pcolormesh(X, Y, np.abs(symbol_vals), shading='auto') plt.colorbar(label='|Symbol|') plt.xlabel('x') plt.ylabel('y') plt.title(f'Symbol Amplitude at ξ={xi0}, η={eta0}') plt.show() def visualize_phase(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Plot the phase (argument) of the pseudodifferential operator's symbol p(x, ξ) or p(x, y, ξ, η). This visualization helps in understanding the oscillatory behavior and regularity properties of the operator in phase space. The phase is displayed modulo 2π using a cyclic colormap ('twilight') to emphasize its periodic nature. Parameters ---------- x_grid : ndarray 1D array of spatial coordinates (x). xi_grid : ndarray 1D array of frequency coordinates (ξ). y_grid : ndarray, optional 2D spatial grid for y-coordinate (in 2D problems). Default is None. eta_grid : ndarray, optional 2D frequency grid for η (in 2D problems). Not used directly but kept for API consistency. xi0 : float, optional Fixed value of ξ for slicing in 2D visualization. Default is 0.0. eta0 : float, optional Fixed value of η for slicing in 2D visualization. Default is 0.0. Notes: - In 1D: Displays arg(p(x, ξ)) over the (x, ξ) phase plane. - In 2D: Displays arg(p(x, y, ξ₀, η₀)) for fixed frequency values (ξ₀, η₀). - Uses plt.pcolormesh with 'twilight' colormap to represent angles from -π to π. Raises: - NotImplementedError: If the spatial dimension is not 1D or 2D. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.pcolormesh(X, XI, np.angle(symbol_vals), shading='auto', cmap='twilight') plt.colorbar(label='arg(Symbol) [rad]') plt.xlabel('x') plt.ylabel('ξ') plt.title('Phase Portrait (arg p(x, ξ))') plt.show() elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.pcolormesh(X, Y, np.angle(symbol_vals), shading='auto', cmap='twilight') plt.colorbar(label='arg(Symbol) [rad]') plt.xlabel('x') plt.ylabel('y') plt.title(f'Phase Portrait at ξ={xi0}, η={eta0}') plt.show() def visualize_characteristic_set(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0, levels=[1e-1]): """ Visualize the characteristic set of the pseudo-differential symbol, defined as the approximate zero set p(x, ξ) ≈ 0. In microlocal analysis, the characteristic set is the locus of points in phase space (x, ξ) where the symbol p(x, ξ) vanishes, playing a key role in understanding propagation of singularities and wavefronts. Parameters ---------- x_grid : ndarray Spatial grid values (1D array) for plotting in 1D or evaluation point in 2D. xi_grid : ndarray Frequency variable grid values (1D array) used to construct the frequency domain. x0 : float, optional Fixed spatial coordinate in 2D case for evaluating the symbol at a specific x position. y0 : float, optional Fixed spatial coordinate in 2D case for evaluating the symbol at a specific y position. Notes ----- - For 1D, this method plots the contour of |p(x, ξ)| = ε with ε = 1e-5 over the (x, ξ) plane. - For 2D, it evaluates the symbol at fixed (x₀, y₀) and plots the characteristic set in the (ξ, η) frequency plane. - This visualization helps identify directions of degeneracy or hypoellipticity of the operator. Raises ------ NotImplementedError If called on a solver with dimensionality other than 1D or 2D. Displays ------ A matplotlib contour plot showing either: - The characteristic curve in the (x, ξ) phase plane (1D), - The characteristic surface slice in the (ξ, η) frequency plane at (x₀, y₀) (2D). """ if self.dim == 1: x_grid = np.asarray(x_grid) xi_grid = np.asarray(xi_grid) X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.contour(X, XI, np.abs(symbol_vals), levels=levels, colors='red') plt.xlabel('x') plt.ylabel('ξ') plt.title('Characteristic Set (p(x, ξ) ≈ 0)') plt.grid(True) plt.show() elif self.dim == 2: if eta_grid is None: raise ValueError("eta_grid must be provided for 2D visualization.") xi_grid = np.asarray(xi_grid) eta_grid = np.asarray(eta_grid) xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij') symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) plt.contour(xi_grid, eta_grid, np.abs(symbol_vals), levels=levels, colors='red') plt.xlabel('ξ') plt.ylabel('η') plt.title(f'Characteristic Set at x={x0}, y={y0}') plt.grid(True) plt.show() else: raise NotImplementedError("Only 1D/2D characteristic sets supported.") def visualize_characteristic_gradient(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0): """ Visualize the norm of the gradient of the symbol in phase space. This method computes the magnitude of the gradient |∇p| of a pseudo-differential symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. The resulting colormap reveals regions where the symbol varies rapidly or remains nearly stationary, which is particularly useful for analyzing characteristic sets. Parameters ---------- x_grid : numpy.ndarray 1D array of spatial coordinates for the x-direction. xi_grid : numpy.ndarray 1D array of frequency coordinates (ξ). y_grid : numpy.ndarray, optional 1D array of spatial coordinates for the y-direction (used in 2D mode). Default is None. eta_grid : numpy.ndarray, optional 1D array of frequency coordinates (η) for the 2D case. Default is None. x0 : float, optional Fixed x-coordinate for evaluating the symbol in 2D. Default is 0.0. y0 : float, optional Fixed y-coordinate for evaluating the symbol in 2D. Default is 0.0. Returns ------- None Displays a 2D colormap of |∇p| over the relevant phase-space domain. Notes ----- - In 1D, the full gradient ∇p = (∂ₓp, ∂ξp) is computed over the (x, ξ) grid. - In 2D, the gradient ∇p = (∂ξp, ∂ηp) is computed at a fixed spatial point (x₀, y₀) over the (ξ, η) grid. - Numerical differentiation is performed using `np.gradient`. - High values of |∇p| indicate rapid variation of the symbol, while low values typically suggest characteristic regions. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) grad_x = np.gradient(symbol_vals, axis=0) grad_xi = np.gradient(symbol_vals, axis=1) grad_norm = np.sqrt(grad_x**2 + grad_xi**2) plt.pcolormesh(X, XI, grad_norm, cmap='inferno', shading='auto') plt.colorbar(label='|∇p|') plt.title('Gradient Norm (High Near Zeros)') plt.grid(True) plt.show() elif self.dim == 2: xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij') symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) grad_xi = np.gradient(symbol_vals, axis=0) grad_eta = np.gradient(symbol_vals, axis=1) grad_norm = np.sqrt(grad_xi**2 + grad_eta**2) plt.pcolormesh(xi_grid, eta_grid, grad_norm, cmap='inferno', shading='auto') plt.colorbar(label='|∇p|') plt.title(f'Gradient Norm at x={x0}, y={y0}') plt.grid(True) plt.show() def visualize_dynamic_wavefront(self, x_grid, t_grid, y_grid=None, xi0=5.0, eta0=0.0): """ Create an animation of a monochromatic wave evolving under the pseudo-differential operator. This method visualizes the time evolution of a wavefront governed by the symbol of the pseudo-differential operator. The wave is initialized as a monochromatic solution with fixed spatial frequencies and evolves according to the dispersion relation ω = p(x, ξ), where p is the symbol of the operator and (ξ, η) are spatial frequencies. Parameters ---------- x_grid : array_like 1D array representing the spatial grid in the x-direction. t_grid : array_like 1D array of time points used to generate the animation frames. y_grid : array_like, optional 1D array representing the spatial grid in the y-direction (required for 2D operators). xi0 : float, optional Initial spatial frequency in the x-direction. Default is 5.0. eta0 : float, optional Initial spatial frequency in the y-direction (used in 2D). Default is 0.0. Returns ------- matplotlib.animation.FuncAnimation Animation object showing the time evolution of the monochromatic wave. Notes ----- - In 1D, visualizes the wave u(x, t) = cos(ξ₀·x − ω·t). - In 2D, visualizes u(x, y, t) = cos(ξ₀·x + η₀·y − ω·t). - The frequency ω is computed as the symbol evaluated at the fixed frequency components. - If the symbol depends on space, it is evaluated at the origin (x=0, y=0). Raises ------ ValueError If `y_grid` is not provided while the operator is 2D. NotImplementedError If the spatial dimension of the operator is neither 1 nor 2. """ if self.dim == 1: # Calculer omega a partir du symbole from sympy import symbols, N x = self.vars_x[0] xi = symbols('xi', real=True) try: omega_symbolic = self.expr.subs({x: 0, xi: xi0}) except: omega_symbolic = self.expr.subs(xi, xi0) omega_val = float(N(omega_symbolic.as_real_imag()[0])) # Preparer les donnees pour l'animation X, T = np.meshgrid(x_grid, t_grid, indexing='ij') # Initialiser la figure et l'axe fig, ax = plt.subplots() ax.set_xlim(x_grid.min(), x_grid.max()) ax.set_ylim(-1.1, 1.1) ax.set_xlabel('x') ax.set_ylabel('u(x, t)') ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t)') ax.grid(True) line, = ax.plot([], [], lw=2) # Pre-calculer les temps T_vals = t_grid def animate(frame): t = T_vals[frame] if frame < len(T_vals) else T_vals[-1] U = np.cos(xi0 * x_grid - omega_val * t) line.set_data(x_grid, U) ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t) at t={t:.2f}') return line, ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=True, repeat=True) plt.show() return ani elif self.dim == 2: # Calculer omega a partir du symbole from sympy import symbols, N x, y = self.vars_x xi, eta = symbols('xi eta', real=True) try: omega_symbolic = self.expr.subs({x: 0, y: 0, xi: xi0, eta: eta0}) except: omega_symbolic = self.expr.subs({xi: xi0, eta: eta0}) omega_val = float(N(omega_symbolic.as_real_imag()[0])) if y_grid is None: raise ValueError("y_grid must be provided for 2D visualization.") # Creer les grilles X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') # Initialiser la figure fig, ax = plt.subplots() ax.set_xlabel('x') ax.set_ylabel('y') # Premiere frame t0 = t_grid[0] if len(t_grid) > 0 else 0.0 U = np.cos(xi0 * X + eta0 * Y - omega_val * t0) extent = [x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()] im = ax.imshow(U, extent=extent, aspect='equal', origin='lower', cmap='seismic', vmin=-1, vmax=1) ax.set_title(f'Dynamic 2D Wavefront at t={t0:.2f}') fig.colorbar(im, ax=ax, label='u(x, y, t)') # Pre-calculer les temps T_vals = t_grid def animate(frame): t = T_vals[frame] if frame < len(T_vals) else T_vals[-1] U = np.cos(xi0 * X + eta0 * Y - omega_val * t) im.set_array(U) ax.set_title(f'Dynamic 2D Wavefront at t={t:.2f}') return [im] ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=False, repeat=True) plt.show() return ani def simulate_evolution(self, x_grid, t_grid, y_grid=None, initial_condition=None, initial_velocity=None, solver_params=None, component='real'): """ Simulate and animate the time evolution of an initial condition under a pseudo-differential operator. This method solves the PDE governed by the action of a pseudo-differential operator `p(x, D)`, either as a first-order or second-order time evolution, and generates an animation of the resulting wave propagation. The order of the PDE depends on whether an initial velocity is provided. Parameters ---------- x_grid : numpy.ndarray Spatial grid in the x-direction. t_grid : numpy.ndarray Temporal grid for the simulation. y_grid : numpy.ndarray, optional Spatial grid in the y-direction (used for 2D problems). initial_condition : callable, optional Function specifying the initial condition u₀(x) or u₀(x, y). initial_velocity : callable, optional Function specifying the initial velocity ∂ₜu₀(x) or ∂ₜu₀(x, y). If provided, a second-order PDE is solved. solver_params : dict, optional Additional keyword arguments passed to the PDE solver. component : {'real', 'imag', 'abs', 'angle'} Specifies which component of the solution to animate. Returns ------- matplotlib.animation.FuncAnimation Animation object showing the evolution of the solution over time. Notes ----- - Solves the PDE: - First-order: ∂ₜu = -i ⋅ p(x, D) u - Second-order: ∂²ₜu = -p(x, D)² u - Supports 1D and 2D simulations depending on the presence of `y_grid`. - The solution is animated using the specified component view. - Useful for visualizing wave propagation and dispersive effects driven by pseudo-differential operators. """ if solver_params is None: solver_params = {} # --- 1. Symbolic variables --- t = symbols('t', real=True) u_sym = Function('u') is_second_order = initial_velocity is not None if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) u = u_sym(t, x) if is_second_order: eq = Eq(diff(u, t, 2), psiOp(self.expr, u)) else: eq = Eq(diff(u, t), psiOp(self.expr, u)) elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) u = u_sym(t, x, y) if is_second_order: eq = Eq(diff(u, t, 2), psiOp(self.expr, u)) else: eq = Eq(diff(u, t), psiOp(self.expr, u)) else: raise NotImplementedError("Only 1D and 2D are supported.") # --- 2. Create the solver --- solver = PDESolver(eq) params = { 'Lx': x_grid.max() - x_grid.min(), 'Nx': len(x_grid), 'Lt': t_grid.max() - t_grid.min(), 'Nt': len(t_grid), 'boundary_condition': 'periodic', 'n_frames': min(100, len(t_grid)) } if self.dim == 2: params['Ly'] = y_grid.max() - y_grid.min() params['Ny'] = len(y_grid) params.update(solver_params) # --- 3. Initial condition --- if initial_condition is None: raise ValueError("initial_condition is None. Please provide a function u₀(x) or u₀(x, y) as the initial condition.") params['initial_condition'] = initial_condition if is_second_order: params['initial_velocity'] = initial_velocity # --- 4. Solving --- print("⚙️ Solving the evolution equation (order {} in time)...".format(2 if is_second_order else 1)) solver.setup(**params) solver.solve() print("✅ Solving completed.") # --- 5. Animation --- print("🎞️ Creating the animation...") ani = solver.animate(component=component) return ani def plot_hamiltonian_flow(self, x0=0.0, xi0=5.0, y0=0.0, eta0=0.0, tmax=1.0, n_steps=100, show_field=True): """ Integrate and plot the Hamiltonian trajectories of the symbol in phase space. This method numerically integrates the Hamiltonian vector field derived from the operator's symbol to visualize how singularities propagate under the flow. It supports both 1D and 2D problems. Parameters ---------- x0, xi0 : float Initial position and frequency (momentum) in 1D. y0, eta0 : float, optional Initial position and frequency in 2D; defaults to zero. tmax : float Final integration time for the ODE solver. n_steps : int Number of time steps used in the integration. Notes ----- - The Hamiltonian vector field is obtained from the symplectic flow of the symbol. - If the field is complex-valued, only its real part is used for integration. - In 1D, the trajectory is plotted in (x, ξ) phase space. - In 2D, the spatial trajectory (x(t), y(t)) is shown along with instantaneous momentum vectors (ξ(t), η(t)) using a quiver plot. Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. Displays -------- matplotlib plot Phase space trajectory(ies) showing the evolution of position and momentum under the Hamiltonian dynamics. """ def make_real(expr): from sympy import re, simplify expr = expr.doit(deep=True) return simplify(re(expr)) H = self.symplectic_flow() if any(im(H[k]) != 0 for k in H): print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.") if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) dxdt_expr = make_real(H['dx/dt']) dxidt_expr = make_real(H['dxi/dt']) dxdt = lambdify((x, xi), dxdt_expr, 'numpy') dxidt = lambdify((x, xi), dxidt_expr, 'numpy') def hamilton(t, Y): x, xi = Y return [dxdt(x, xi), dxidt(x, xi)] sol = solve_ivp(hamilton, [0, tmax], [x0, xi0], t_eval=np.linspace(0, tmax, n_steps)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_steps: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_steps = n_points x_vals, xi_vals = sol.y plt.plot(x_vals, xi_vals) plt.xlabel("x") plt.ylabel("ξ") plt.title("Hamiltonian Flow in Phase Space (1D)") plt.grid(True) plt.show() elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy') dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy') dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy') detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy') def hamilton(t, Y): x, y, xi, eta = Y return [ dxdt(x, y, xi, eta), dydt(x, y, xi, eta), dxidt(x, y, xi, eta), detadt(x, y, xi, eta) ] sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0], t_eval=np.linspace(0, tmax, n_steps)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_steps: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_steps = n_points x_vals, y_vals, xi_vals, eta_vals = sol.y plt.plot(x_vals, y_vals, label='Position') plt.quiver(x_vals, y_vals, xi_vals, eta_vals, scale=20, width=0.003, alpha=0.5, color='r') # Vector field of the flow (optional) if show_field: X, Y = np.meshgrid(np.linspace(min(x_vals), max(x_vals), 20), np.linspace(min(y_vals), max(y_vals), 20)) XI, ETA = xi0 * np.ones_like(X), eta0 * np.ones_like(Y) U = dxdt(X, Y, XI, ETA) V = dydt(X, Y, XI, ETA) plt.quiver(X, Y, U, V, color='gray', alpha=0.2, scale=30, width=0.002) plt.xlabel("x") plt.ylabel("y") plt.title("Hamiltonian Flow in Phase Space (2D)") plt.legend() plt.grid(True) plt.axis('equal') plt.show() def plot_symplectic_vector_field(self, xlim=(-2, 2), klim=(-5, 5), density=30): """ Visualize the symplectic vector field (Hamiltonian vector field) associated with the operator's symbol. The plotted vector field corresponds to (∂_ξ p, -∂_x p), where p(x, ξ) is the principal symbol of the pseudo-differential operator. This field governs the bicharacteristic flow in phase space. Parameters ---------- xlim : tuple of float Range for spatial variable x, as (x_min, x_max). klim : tuple of float Range for frequency variable ξ, as (ξ_min, ξ_max). density : int Number of grid points per axis for the visualization grid. Raises ------ NotImplementedError If called on a 2D operator (currently only 1D implementation available). Notes ----- - Only supports one-dimensional operators. - Uses symbolic differentiation to compute ∂_ξ p and ∂_x p. - Numerical evaluation is done via lambdify with NumPy backend. - Visualization uses matplotlib quiver plot to show vector directions. """ x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') if self.dim != 1: raise NotImplementedError("Only 1D version implemented.") x, = self.vars_x xi = symbols('xi', real=True) H = self.symplectic_flow() dxdt = lambdify((x, xi), simplify(H['dx/dt']), 'numpy') dxidt = lambdify((x, xi), simplify(H['dxi/dt']), 'numpy') U = dxdt(X, XI) V = dxidt(X, XI) plt.quiver(X, XI, U, V, scale=10, width=0.005) plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Symplectic Vector Field (1D)") plt.grid(True) plt.show() def visualize_micro_support(self, xlim=(-2, 2), klim=(-10, 10), threshold=1e-3, density=300): """ Visualize the micro-support of the operator by plotting the inverse of the symbol magnitude 1 / |p(x, ξ)|. The micro-support provides insight into the singularities of a pseudo-differential operator in phase space (x, ξ). Regions where |p(x, ξ)| is small correspond to large values in 1/|p(x, ξ)|, highlighting areas of significant operator influence or singularity. Parameters ---------- xlim : tuple Spatial domain limits (x_min, x_max). klim : tuple Frequency domain limits (ξ_min, ξ_max). threshold : float Threshold below which |p(x, ξ)| is considered effectively zero; used for numerical stability. density : int Number of grid points along each axis for visualization resolution. Raises ------ NotImplementedError If called on a solver with dimension greater than 1 (only 1D visualization is supported). Notes ----- - This method evaluates the symbol p(x, ξ) over a grid and plots its reciprocal to emphasize regions where the symbol is near zero. - A small constant (1e-10) is added to the denominator to avoid division by zero. - The resulting plot helps identify characteristic sets and wavefront set approximations. """ if self.dim != 1: raise NotImplementedError("Only 1D micro-support visualization implemented.") x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') Z = np.abs(self.p_func(X, XI)) plt.contourf(X, XI, 1 / (Z + 1e-10), levels=100, cmap='inferno') plt.colorbar(label=r'$1/|p(x,\xi)|$') plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Micro-Support Estimate (1/|Symbol|)") plt.show() def group_velocity_field(self, xlim=(-2, 2), klim=(-10, 10), density=30): """ Plot the group velocity field ∇_ξ p(x, ξ) for 1D pseudo-differential operators. The group velocity represents the speed at which waves of different frequencies propagate in a dispersive medium. It is defined as the gradient of the symbol p(x, ξ) with respect to the frequency variable ξ. Parameters ---------- xlim : tuple of float Spatial domain limits (x-axis). klim : tuple of float Frequency domain limits (ξ-axis). density : int Number of grid points per axis used for visualization. Raises ------ NotImplementedError If called on a 2D operator, since this visualization is only implemented for 1D. Notes ----- - This method visualizes the vector field (∂p/∂ξ) in phase space. - Used for analyzing wave propagation properties and dispersion relations. - Requires symbolic expression self.expr depending on x and ξ. """ if self.dim != 1: raise NotImplementedError("Only 1D group velocity visualization implemented.") x, = self.vars_x xi = symbols('xi', real=True) dp_dxi = diff(self.expr, xi) grad_func = lambdify((x, xi), dp_dxi, 'numpy') x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') V = grad_func(X, XI) plt.quiver(X, XI, np.ones_like(V), V, scale=10, width=0.004) plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Group Velocity Field (1D)") plt.grid(True) plt.show() def animate_singularity(self, xi0=5.0, eta0=0.0, x0=0.0, y0=0.0, tmax=4.0, n_frames=100, projection=None): """ Animate the propagation of a singularity under the Hamiltonian flow. This method visualizes how a singularity (x₀, y₀, ξ₀, η₀) evolves in phase space according to the Hamiltonian dynamics induced by the principal symbol of the operator. The animation integrates the Hamiltonian equations of motion and supports various projections: position (x-y), frequency (ξ-η), or mixed phase space coordinates. Parameters ---------- xi0, eta0 : float Initial frequency components (ξ₀, η₀). x0, y0 : float Initial spatial coordinates (x₀, y₀). tmax : float Total time of integration (final animation time). n_frames : int Number of frames in the resulting animation. projection : str or None Type of projection to display: - 'position' : x vs y (or x alone in 1D) - 'frequency': ξ vs η (or ξ alone in 1D) - 'phase' : mixed coordinates like x vs ξ or x vs η If None, defaults to 'phase' in 1D and 'position' in 2D. Returns ------- matplotlib.animation.FuncAnimation Animation object that can be displayed interactively in Jupyter notebooks or saved as a video. Notes ----- - In 1D, only one spatial and one frequency variable are used. - Complex-valued Hamiltonian fields are truncated to their real parts for integration. - Trajectories are shown with both instantaneous position (dot) and full path (dashed line). """ rc('animation', html='jshtml') def make_real(expr): from sympy import re, simplify expr = expr.doit(deep=True) return simplify(re(expr)) H = self.symplectic_flow() H = {k: v.doit(deep=True) for k, v in H.items()} print("H = ", H) if any(im(H[k]) != 0 for k in H): print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.") if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) dxdt = lambdify((x, xi), make_real(H['dx/dt']), 'numpy') dxidt = lambdify((x, xi), make_real(H['dxi/dt']), 'numpy') def hamilton(t, Y): x, xi = Y return [dxdt(x, xi), dxidt(x, xi)] sol = solve_ivp(hamilton, [0, tmax], [x0, xi0], t_eval=np.linspace(0, tmax, n_frames)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_frames: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_frames = n_points x_vals, xi_vals = sol.y if projection is None: projection = 'phase' fig, ax = plt.subplots() point, = ax.plot([], [], 'ro') traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5) if projection == 'phase': ax.set_xlabel('x') ax.set_ylabel(r'$\xi$') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) def update(i): point.set_data([x_vals[i]], [xi_vals[i]]) traj.set_data(x_vals[:i+1], xi_vals[:i+1]) return point, traj elif projection == 'position': ax.set_xlabel('x') ax.set_ylabel('x') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(x_vals) - 1, np.max(x_vals) + 1) def update(i): point.set_data([x_vals[i]], [x_vals[i]]) traj.set_data(x_vals[:i+1], x_vals[:i+1]) return point, traj elif projection == 'frequency': ax.set_xlabel(r'$\xi$') ax.set_ylabel(r'$\xi$') ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) def update(i): point.set_data([xi_vals[i]], [xi_vals[i]]) traj.set_data(xi_vals[:i+1], xi_vals[:i+1]) return point, traj else: raise ValueError("Invalid projection mode") ax.set_title(f"1D Singularity Flow ({projection})") ax.grid(True) ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50) plt.close(fig) return ani elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy') dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy') dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy') detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy') def hamilton(t, Y): x, y, xi, eta = Y return [ dxdt(x, y, xi, eta), dydt(x, y, xi, eta), dxidt(x, y, xi, eta), detadt(x, y, xi, eta) ] sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0], t_eval=np.linspace(0, tmax, n_frames)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_frames: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_frames = n_points x_vals, y_vals, xi_vals, eta_vals = sol.y if projection is None: projection = 'position' fig, ax = plt.subplots() point, = ax.plot([], [], 'ro') traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5) if projection == 'position': ax.set_xlabel('x') ax.set_ylabel('y') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(y_vals) - 1, np.max(y_vals) + 1) def update(i): point.set_data([x_vals[i]], [y_vals[i]]) traj.set_data(x_vals[:i+1], y_vals[:i+1]) return point, traj elif projection == 'frequency': ax.set_xlabel(r'$\xi$') ax.set_ylabel(r'$\eta$') ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1) def update(i): point.set_data([xi_vals[i]], [eta_vals[i]]) traj.set_data(xi_vals[:i+1], eta_vals[:i+1]) return point, traj elif projection == 'phase': ax.set_xlabel('x') ax.set_ylabel(r'$\eta$') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1) def update(i): point.set_data([x_vals[i]], [eta_vals[i]]) traj.set_data(x_vals[:i+1], eta_vals[:i+1]) return point, traj else: raise ValueError("Invalid projection mode") ax.set_title(f"2D Singularity Flow ({projection})") ax.grid(True) ax.axis('equal') ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50) plt.close(fig) return ani def interactive_symbol_analysis(pseudo_op, xlim=(-2, 2), ylim=(-2, 2), xi_range=(0.1, 5), eta_range=(-5, 5), density=100): """ Launch an interactive dashboard for symbol exploration using ipywidgets. This function provides a user-friendly interface to visualize various aspects of the pseudo-differential operator's symbol. It supports multiple visualization modes in both 1D and 2D, including group velocity fields, micro-support estimates, symplectic vector fields, symbol amplitude/phase, cotangent fiber structure, characteristic sets, wavefront sets, and Hamiltonian flows. Parameters ---------- pseudo_op : PseudoDifferentialOperator The pseudo-differential operator whose symbol is to be analyzed interactively. xlim, ylim : tuple of float Spatial domain limits along x and y axes respectively. xi_range, eta_range : tuple Frequency domain limits along ξ and η axes respectively. density : int Number of points per axis used to construct the evaluation grid. Controls resolution. Notes ----- - In 1D mode, sliders control the fixed frequency (ξ₀) and spatial position (x₀). - In 2D mode, additional sliders control the second frequency component (η₀) and second spatial coordinate (y₀). - Visualization updates dynamically as parameters are adjusted via sliders or dropdown menus. - Supported visualization modes: 'Group Velocity Field' : ∇_ξ p(x,ξ) or ∇_{ξ,η} p(x,y,ξ,η) 'Micro-Support (1/|p|)' : Reciprocal of symbol magnitude 'Symplectic Vector Field' : (∇_ξ p, -∇_x p) or similar in 2D 'Symbol Amplitude' : |p(x,ξ)| or |p(x,y,ξ,η)| 'Symbol Phase' : arg(p(x,ξ)) or similar in 2D 'Cotangent Fiber' : Structure of symbol over frequency space at fixed x 'Characteristic Set' : Zero set approximation {p ≈ 0} 'Characteristic Gradient' : |∇p(x, ξ)| or |∇p(x₀, y₀, ξ, η)| 'Wavefront Set' : High-frequency singularities detected via symbol interaction 'Hamiltonian Flow' : Trajectories generated by the Hamiltonian vector field Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. Prints ------ Interactive matplotlib figures with dynamic updates based on widget inputs. """ dim = pseudo_op.dim expr = pseudo_op.expr vars_x = pseudo_op.vars_x mode_selector = Dropdown( options=[ 'Group Velocity Field', 'Micro-Support (1/|p|)', 'Symplectic Vector Field', 'Symbol Amplitude', 'Symbol Phase', 'Cotangent Fiber', 'Characteristic Set', 'Characteristic Gradient', 'Wavefront Set', 'Hamiltonian Flow', ], value='Group Velocity Field', description='Mode:' ) x_vals = np.linspace(*xlim, density) if dim == 2: y_vals = np.linspace(*ylim, density) if dim == 1: x, = vars_x xi = symbols('xi', real=True) grad_func = lambdify((x, xi), diff(expr, xi), 'numpy') symplectic_func = lambdify((x, xi), [diff(expr, xi), -diff(expr, x)], 'numpy') symbol_func = lambdify((x, xi), expr, 'numpy') def plot_1d(mode, xi0, x0): X = x_vals[:, None] if mode == 'Group Velocity Field': V = grad_func(X, xi0) plt.quiver(X, V, np.ones_like(V), V, scale=10, width=0.004) plt.title(f'Group Velocity Field at ξ={xi0:.2f}') elif mode == 'Micro-Support (1/|p|)': Z = 1 / (np.abs(symbol_func(X, xi0)) + 1e-10) plt.plot(x_vals, Z) plt.title(f'Micro-Support (1/|p|) at ξ={xi0:.2f}') elif mode == 'Symplectic Vector Field': U, V = symplectic_func(X, xi0) plt.quiver(X, V, U, V, scale=10, width=0.004) plt.title(f'Symplectic Field at ξ={xi0:.2f}') elif mode == 'Symbol Amplitude': Z = np.abs(symbol_func(X, xi0)) plt.plot(x_vals, Z) plt.title(f'Symbol Amplitude |p(x,ξ)| at ξ={xi0:.2f}') elif mode == 'Symbol Phase': Z = np.angle(symbol_func(X, xi0)) plt.plot(x_vals, Z) plt.title(f'Symbol Phase arg(p(x,ξ)) at ξ={xi0:.2f}') elif mode == 'Cotangent Fiber': pseudo_op.visualize_fiber(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Characteristic Set': pseudo_op.visualize_characteristic_set(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Characteristic Gradient': pseudo_op.visualize_characteristic_gradient(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Wavefront Set': pseudo_op.visualize_wavefront(x_vals, np.linspace(*xi_range, density), xi0=xi0) elif mode == 'Hamiltonian Flow': pseudo_op.plot_hamiltonian_flow(x0=x0, xi0=xi0) interact(plot_1d, mode=mode_selector, xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'), x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀')) elif dim == 2: x, y = vars_x xi, eta = symbols('xi eta', real=True) grad_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy') symplectic_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy') symbol_func = lambdify((x, y, xi, eta), expr, 'numpy') def plot_2d(mode, xi0, eta0, x0, y0): X, Y = np.meshgrid(x_vals, y_vals, indexing='ij') if mode == 'Group Velocity Field': U, V = grad_func(X, Y, xi0, eta0) plt.quiver(X, Y, U, V, scale=10, width=0.004) plt.title(f'Group Velocity Field at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Micro-Support (1/|p|)': Z = 1 / (np.abs(symbol_func(X, Y, xi0, eta0)) + 1e-10) plt.pcolormesh(X, Y, Z, shading='auto', cmap='inferno') plt.colorbar(label='1/|p|') plt.title(f'Micro-Support at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symplectic Vector Field': U, V = symplectic_func(X, Y, xi0, eta0) plt.quiver(X, Y, U, V, scale=10, width=0.004) plt.title(f'Symplectic Field at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symbol Amplitude': Z = np.abs(symbol_func(X, Y, xi0, eta0)) plt.pcolormesh(X, Y, Z, shading='auto') plt.colorbar(label='|p(x,y,ξ,η)|') plt.title(f'Symbol Amplitude at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symbol Phase': Z = np.angle(symbol_func(X, Y, xi0, eta0)) plt.pcolormesh(X, Y, Z, shading='auto', cmap='twilight') plt.colorbar(label='arg(p)') plt.title(f'Symbol Phase at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Cotangent Fiber': pseudo_op.visualize_fiber(np.linspace(*xi_range, density), np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Characteristic Set': pseudo_op.visualize_characteristic_set(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Characteristic Gradient': pseudo_op.visualize_characteristic_gradient(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Wavefront Set': pseudo_op.visualize_wavefront(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), xi0=xi0, eta0=eta0) elif mode == 'Hamiltonian Flow': pseudo_op.plot_hamiltonian_flow(x0=x0, y0=y0, xi0=xi0, eta0=eta0) interact(plot_2d, mode=mode_selector, xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'), eta0=FloatSlider(min=eta_range[0], max=eta_range[1], step=0.1, value=1.0, description='η₀'), x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀'), y0=FloatSlider(min=ylim[0], max=ylim[1], step=0.1, value=0.0, description='y₀'))Methods
def animate_singularity(self, xi0=5.0, eta0=0.0, x0=0.0, y0=0.0, tmax=4.0, n_frames=100, projection=None)-
Animate the propagation of a singularity under the Hamiltonian flow.
This method visualizes how a singularity (x₀, y₀, ξ₀, η₀) evolves in phase space according to the Hamiltonian dynamics induced by the principal symbol of the operator. The animation integrates the Hamiltonian equations of motion and supports various projections: position (x-y), frequency (ξ-η), or mixed phase space coordinates.
Parameters
xi0,eta0:float- Initial frequency components (ξ₀, η₀).
x0,y0:float- Initial spatial coordinates (x₀, y₀).
tmax:float- Total time of integration (final animation time).
n_frames:int- Number of frames in the resulting animation.
projection:strorNone- Type of projection to display: - 'position' : x vs y (or x alone in 1D) - 'frequency': ξ vs η (or ξ alone in 1D) - 'phase' : mixed coordinates like x vs ξ or x vs η If None, defaults to 'phase' in 1D and 'position' in 2D.
Returns
matplotlib.animation.FuncAnimation- Animation object that can be displayed interactively in Jupyter notebooks or saved as a video.
Notes
- In 1D, only one spatial and one frequency variable are used.
- Complex-valued Hamiltonian fields are truncated to their real parts for integration.
- Trajectories are shown with both instantaneous position (dot) and full path (dashed line).
Expand source code
def animate_singularity(self, xi0=5.0, eta0=0.0, x0=0.0, y0=0.0, tmax=4.0, n_frames=100, projection=None): """ Animate the propagation of a singularity under the Hamiltonian flow. This method visualizes how a singularity (x₀, y₀, ξ₀, η₀) evolves in phase space according to the Hamiltonian dynamics induced by the principal symbol of the operator. The animation integrates the Hamiltonian equations of motion and supports various projections: position (x-y), frequency (ξ-η), or mixed phase space coordinates. Parameters ---------- xi0, eta0 : float Initial frequency components (ξ₀, η₀). x0, y0 : float Initial spatial coordinates (x₀, y₀). tmax : float Total time of integration (final animation time). n_frames : int Number of frames in the resulting animation. projection : str or None Type of projection to display: - 'position' : x vs y (or x alone in 1D) - 'frequency': ξ vs η (or ξ alone in 1D) - 'phase' : mixed coordinates like x vs ξ or x vs η If None, defaults to 'phase' in 1D and 'position' in 2D. Returns ------- matplotlib.animation.FuncAnimation Animation object that can be displayed interactively in Jupyter notebooks or saved as a video. Notes ----- - In 1D, only one spatial and one frequency variable are used. - Complex-valued Hamiltonian fields are truncated to their real parts for integration. - Trajectories are shown with both instantaneous position (dot) and full path (dashed line). """ rc('animation', html='jshtml') def make_real(expr): from sympy import re, simplify expr = expr.doit(deep=True) return simplify(re(expr)) H = self.symplectic_flow() H = {k: v.doit(deep=True) for k, v in H.items()} print("H = ", H) if any(im(H[k]) != 0 for k in H): print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.") if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) dxdt = lambdify((x, xi), make_real(H['dx/dt']), 'numpy') dxidt = lambdify((x, xi), make_real(H['dxi/dt']), 'numpy') def hamilton(t, Y): x, xi = Y return [dxdt(x, xi), dxidt(x, xi)] sol = solve_ivp(hamilton, [0, tmax], [x0, xi0], t_eval=np.linspace(0, tmax, n_frames)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_frames: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_frames = n_points x_vals, xi_vals = sol.y if projection is None: projection = 'phase' fig, ax = plt.subplots() point, = ax.plot([], [], 'ro') traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5) if projection == 'phase': ax.set_xlabel('x') ax.set_ylabel(r'$\xi$') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) def update(i): point.set_data([x_vals[i]], [xi_vals[i]]) traj.set_data(x_vals[:i+1], xi_vals[:i+1]) return point, traj elif projection == 'position': ax.set_xlabel('x') ax.set_ylabel('x') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(x_vals) - 1, np.max(x_vals) + 1) def update(i): point.set_data([x_vals[i]], [x_vals[i]]) traj.set_data(x_vals[:i+1], x_vals[:i+1]) return point, traj elif projection == 'frequency': ax.set_xlabel(r'$\xi$') ax.set_ylabel(r'$\xi$') ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) ax.set_ylim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) def update(i): point.set_data([xi_vals[i]], [xi_vals[i]]) traj.set_data(xi_vals[:i+1], xi_vals[:i+1]) return point, traj else: raise ValueError("Invalid projection mode") ax.set_title(f"1D Singularity Flow ({projection})") ax.grid(True) ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50) plt.close(fig) return ani elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy') dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy') dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy') detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy') def hamilton(t, Y): x, y, xi, eta = Y return [ dxdt(x, y, xi, eta), dydt(x, y, xi, eta), dxidt(x, y, xi, eta), detadt(x, y, xi, eta) ] sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0], t_eval=np.linspace(0, tmax, n_frames)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_frames: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_frames = n_points x_vals, y_vals, xi_vals, eta_vals = sol.y if projection is None: projection = 'position' fig, ax = plt.subplots() point, = ax.plot([], [], 'ro') traj, = ax.plot([], [], 'b--', lw=1, alpha=0.5) if projection == 'position': ax.set_xlabel('x') ax.set_ylabel('y') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(y_vals) - 1, np.max(y_vals) + 1) def update(i): point.set_data([x_vals[i]], [y_vals[i]]) traj.set_data(x_vals[:i+1], y_vals[:i+1]) return point, traj elif projection == 'frequency': ax.set_xlabel(r'$\xi$') ax.set_ylabel(r'$\eta$') ax.set_xlim(np.min(xi_vals) - 1, np.max(xi_vals) + 1) ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1) def update(i): point.set_data([xi_vals[i]], [eta_vals[i]]) traj.set_data(xi_vals[:i+1], eta_vals[:i+1]) return point, traj elif projection == 'phase': ax.set_xlabel('x') ax.set_ylabel(r'$\eta$') ax.set_xlim(np.min(x_vals) - 1, np.max(x_vals) + 1) ax.set_ylim(np.min(eta_vals) - 1, np.max(eta_vals) + 1) def update(i): point.set_data([x_vals[i]], [eta_vals[i]]) traj.set_data(x_vals[:i+1], eta_vals[:i+1]) return point, traj else: raise ValueError("Invalid projection mode") ax.set_title(f"2D Singularity Flow ({projection})") ax.grid(True) ax.axis('equal') ani = animation.FuncAnimation(fig, update, frames=n_frames, interval=50) plt.close(fig) return ani def asymptotic_expansion(self, order=3)-
Compute the asymptotic expansion of the symbol as |ξ| → ∞ (high-frequency regime).
This method expands the pseudo-differential symbol in inverse powers of the frequency variable(s), either in 1D or 2D. It handles both polynomial and exponential symbols by performing a series expansion in 1/|ξ| up to the specified order.
The expansion is performed directly in Cartesian coordinates for 1D symbols. For 2D symbols, the method uses polar coordinates (ρ, θ) to perform the expansion at infinity in ρ, then converts the result back to Cartesian coordinates.
Parameters
order:int, optional- Maximum order of the asymptotic expansion. Default is 3.
Returns
sympy.Expr- The asymptotic expansion of the symbol up to the given order, expressed in Cartesian coordinates. If expansion fails, returns the original unexpanded symbol.
Notes: - In 1D: expansion is performed directly in terms of ξ. - In 2D: the symbol is first rewritten in polar coordinates (ρ,θ), expanded asymptotically in ρ → ∞, then converted back to Cartesian coordinates (ξ,η). - Handles special case when the symbol is an exponential function by expanding its argument. - Symbolic normalization is applied early (via
simplify) for 2D expressions to improve convergence. - Robust to failures: catches exceptions and issues warnings instead of raising errors. - Final expression is simplified usingpowdenestandexpandfor improved readability.Expand source code
def asymptotic_expansion(self, order=3): """ Compute the asymptotic expansion of the symbol as |ξ| → ∞ (high-frequency regime). This method expands the pseudo-differential symbol in inverse powers of the frequency variable(s), either in 1D or 2D. It handles both polynomial and exponential symbols by performing a series expansion in 1/|ξ| up to the specified order. The expansion is performed directly in Cartesian coordinates for 1D symbols. For 2D symbols, the method uses polar coordinates (ρ, θ) to perform the expansion at infinity in ρ, then converts the result back to Cartesian coordinates. Parameters ---------- order : int, optional Maximum order of the asymptotic expansion. Default is 3. Returns ------- sympy.Expr The asymptotic expansion of the symbol up to the given order, expressed in Cartesian coordinates. If expansion fails, returns the original unexpanded symbol. Notes: - In 1D: expansion is performed directly in terms of ξ. - In 2D: the symbol is first rewritten in polar coordinates (ρ,θ), expanded asymptotically in ρ → ∞, then converted back to Cartesian coordinates (ξ,η). - Handles special case when the symbol is an exponential function by expanding its argument. - Symbolic normalization is applied early (via `simplify`) for 2D expressions to improve convergence. - Robust to failures: catches exceptions and issues warnings instead of raising errors. - Final expression is simplified using `powdenest` and `expand` for improved readability. """ p = self.expr if self.dim == 1: xi = symbols('xi', real=True, positive=True) try: # Case: exponential function if p.func == exp and len(p.args) == 1: arg = p.args[0] arg_series = series(arg, xi, oo, n=order).removeO() expanded = series(exp(expand(arg_series)), xi, oo, n=order).removeO() return simplify(powdenest(expanded, force=True)) else: expanded = series(p, xi, oo, n=order).removeO() return simplify(powdenest(expanded, force=True)) except Exception as e: print(f"Warning: 1D expansion failed: {e}") return p elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) rho, theta = symbols('rho theta', real=True, positive=True) # Normalize before substitution p = simplify(p) # Substitute polar coordinates p_polar = p.subs({ xi: rho * cos(theta), eta: rho * sin(theta) }) try: # Handle exponentials if p_polar.func == exp and len(p_polar.args) == 1: arg = p_polar.args[0] arg_series = series(arg, rho, oo, n=order).removeO() expanded = series(exp(expand(arg_series)), rho, oo, n=order).removeO() else: expanded = series(p_polar, rho, oo, n=order).removeO() # Convert back to Cartesian norm = sqrt(xi**2 + eta**2) expansion_cart = expanded.subs({ rho: norm, cos(theta): xi / norm, sin(theta): eta / norm }) # Final simplifications result = simplify(powdenest(expansion_cart, force=True)) result = expand(result) return result except Exception as e: print(f"Warning: 2D expansion failed: {e}") return p def clear_cache(self)-
Clear cached symbol evaluations.
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def clear_cache(self): """ Clear cached symbol evaluations. """ self.symbol_cached = None def compose_asymptotic(self, other, order=1)-
Compose this pseudo-differential operator with another using formal asymptotic expansion.
This method computes the composition symbol via an asymptotic expansion in powers of derivatives, following the symbolic calculus of pseudo-differential operators. The composition is performed up to the specified order and respects the dimensionality (1D or 2D) of the operators.
Parameters
other:PseudoDifferentialOperator- The pseudo-differential operator to compose with this one.
order:int, default=1- Maximum order of the asymptotic expansion. Higher values include more terms in the symbolic composition, increasing accuracy at the cost of complexity.
Returns
sympy.Expr- Symbolic expression representing the asymptotic expansion of the composed operator.
Notes
- In 1D, the composition uses the formula: (p ∘ q)(x, ξ) ~ Σₙ (1/n!) ∂_ξⁿ p(x, ξ) ∂_xⁿ q(x, ξ) (i)^{-n}
- In 2D, the multi-index generalization is used: (p ∘ q)(x, y, ξ, η) ~ Σₙ Σᵢ (1/(i! j!)) ∂_ξⁱ∂_ηʲ p ∂_xⁱ∂_yʲ q (i)^{-n}, where n = i + j.
- This expansion is valid for symbols admitting an asymptotic series representation.
- Operators must be defined on the same spatial domain (same dimension).
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def compose_asymptotic(self, other, order=1): """ Compose this pseudo-differential operator with another using formal asymptotic expansion. This method computes the composition symbol via an asymptotic expansion in powers of derivatives, following the symbolic calculus of pseudo-differential operators. The composition is performed up to the specified order and respects the dimensionality (1D or 2D) of the operators. Parameters ---------- other : PseudoDifferentialOperator The pseudo-differential operator to compose with this one. order : int, default=1 Maximum order of the asymptotic expansion. Higher values include more terms in the symbolic composition, increasing accuracy at the cost of complexity. Returns ------- sympy.Expr Symbolic expression representing the asymptotic expansion of the composed operator. Notes ----- - In 1D, the composition uses the formula: (p ∘ q)(x, ξ) ~ Σₙ (1/n!) ∂_ξⁿ p(x, ξ) ∂_xⁿ q(x, ξ) (i)^{-n} - In 2D, the multi-index generalization is used: (p ∘ q)(x, y, ξ, η) ~ Σₙ Σᵢ (1/(i! j!)) ∂_ξⁱ∂_ηʲ p ∂_xⁱ∂_yʲ q (i)^{-n}, where n = i + j. - This expansion is valid for symbols admitting an asymptotic series representation. - Operators must be defined on the same spatial domain (same dimension). """ assert self.dim == other.dim, "Operator dimensions must match" p, q = self.expr, other.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) result = 0 for n in range(order + 1): term = (1 / factorial(n)) * diff(p, xi, n) * diff(q, x, n) * (1j)**(-n) result += term elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) result = 0 for n in range(order + 1): for i in range(n + 1): j = n - i term = (1 / (factorial(i) * factorial(j))) * \ diff(p, xi, i, eta, j) * diff(q, x, i, y, j) * (1j)**(-n) result += term return result def evaluate(self, X, Y, KX, KY, cache=True)-
Evaluate the pseudo-differential operator's symbol on a grid of spatial and frequency coordinates.
The method dynamically selects between 1D and 2D evaluation based on the spatial dimension. If caching is enabled and a cached symbol exists, it returns the cached result to avoid recomputation.
Parameters
X,Y:ndarray- Spatial grid coordinates. In 1D, Y is ignored.
KX,KY:ndarray- Frequency grid coordinates. In 1D, KY is ignored.
cache:bool, default=True- If True, stores the computed symbol for reuse in subsequent calls to avoid redundant computation.
Returns
ndarray- Evaluated symbol values over the input grid. Shape matches the input spatial/frequency grids.
Raises
NotImplementedError- If the spatial dimension is not 1D or 2D.
Expand source code
def evaluate(self, X, Y, KX, KY, cache=True): """ Evaluate the pseudo-differential operator's symbol on a grid of spatial and frequency coordinates. The method dynamically selects between 1D and 2D evaluation based on the spatial dimension. If caching is enabled and a cached symbol exists, it returns the cached result to avoid recomputation. Parameters ---------- X, Y : ndarray Spatial grid coordinates. In 1D, Y is ignored. KX, KY : ndarray Frequency grid coordinates. In 1D, KY is ignored. cache : bool, default=True If True, stores the computed symbol for reuse in subsequent calls to avoid redundant computation. Returns ------- ndarray Evaluated symbol values over the input grid. Shape matches the input spatial/frequency grids. Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. """ if cache and self.symbol_cached is not None: return self.symbol_cached if self.dim == 1: symbol = self.p_func(X, KX) elif self.dim == 2: symbol = self.p_func(X, Y, KX, KY) if cache: self.symbol_cached = symbol return symbol def formal_adjoint(self)-
Compute the formal adjoint symbol P* of the pseudo-differential operator.
The adjoint is defined such that for any test functions u and v, ⟨P u, v⟩ = ⟨u, P* v⟩ holds in the distributional sense. This is obtained by taking the complex conjugate of the symbol and expanding it asymptotically at infinity to ensure proper behavior under integration by parts.
Returns
sympy.Expr- The adjoint symbol P(x, ξ) in 1D or P(x, y, ξ, η) in 2D.
Notes: - In 1D, the expansion is performed in powers of 1/|ξ|. - In 2D, the expansion is radial in |ξ| = sqrt(ξ² + η²). - This method ensures symbolic simplifications for readability and efficiency.
Expand source code
def formal_adjoint(self): """ Compute the formal adjoint symbol P* of the pseudo-differential operator. The adjoint is defined such that for any test functions u and v, ⟨P u, v⟩ = ⟨u, P* v⟩ holds in the distributional sense. This is obtained by taking the complex conjugate of the symbol and expanding it asymptotically at infinity to ensure proper behavior under integration by parts. Returns ------- sympy.Expr The adjoint symbol P*(x, ξ) in 1D or P*(x, y, ξ, η) in 2D. Notes: - In 1D, the expansion is performed in powers of 1/|ξ|. - In 2D, the expansion is radial in |ξ| = sqrt(ξ² + η²). - This method ensures symbolic simplifications for readability and efficiency. """ p = self.expr if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) p_star = conjugate(p) p_star = simplify(series(p_star, xi, oo, n=6).removeO()) return p_star elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) p_star = conjugate(p) p_star = simplify(series(p_star, sqrt(xi**2 + eta**2), oo, n=6).removeO()) return p_star def group_velocity_field(self, xlim=(-2, 2), klim=(-10, 10), density=30)-
Plot the group velocity field ∇_ξ p(x, ξ) for 1D pseudo-differential operators.
The group velocity represents the speed at which waves of different frequencies propagate in a dispersive medium. It is defined as the gradient of the symbol p(x, ξ) with respect to the frequency variable ξ.
Parameters
xlim:tupleoffloat- Spatial domain limits (x-axis).
klim:tupleoffloat- Frequency domain limits (ξ-axis).
density:int- Number of grid points per axis used for visualization.
Raises
NotImplementedError- If called on a 2D operator, since this visualization is only implemented for 1D.
Notes
- This method visualizes the vector field (∂p/∂ξ) in phase space.
- Used for analyzing wave propagation properties and dispersion relations.
- Requires symbolic expression self.expr depending on x and ξ.
Expand source code
def group_velocity_field(self, xlim=(-2, 2), klim=(-10, 10), density=30): """ Plot the group velocity field ∇_ξ p(x, ξ) for 1D pseudo-differential operators. The group velocity represents the speed at which waves of different frequencies propagate in a dispersive medium. It is defined as the gradient of the symbol p(x, ξ) with respect to the frequency variable ξ. Parameters ---------- xlim : tuple of float Spatial domain limits (x-axis). klim : tuple of float Frequency domain limits (ξ-axis). density : int Number of grid points per axis used for visualization. Raises ------ NotImplementedError If called on a 2D operator, since this visualization is only implemented for 1D. Notes ----- - This method visualizes the vector field (∂p/∂ξ) in phase space. - Used for analyzing wave propagation properties and dispersion relations. - Requires symbolic expression self.expr depending on x and ξ. """ if self.dim != 1: raise NotImplementedError("Only 1D group velocity visualization implemented.") x, = self.vars_x xi = symbols('xi', real=True) dp_dxi = diff(self.expr, xi) grad_func = lambdify((x, xi), dp_dxi, 'numpy') x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') V = grad_func(X, XI) plt.quiver(X, XI, np.ones_like(V), V, scale=10, width=0.004) plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Group Velocity Field (1D)") plt.grid(True) plt.show() def interactive_symbol_analysis(pseudo_op, xlim=(-2, 2), ylim=(-2, 2), xi_range=(0.1, 5), eta_range=(-5, 5), density=100)-
Launch an interactive dashboard for symbol exploration using ipywidgets.
This function provides a user-friendly interface to visualize various aspects of the pseudo-differential operator's symbol. It supports multiple visualization modes in both 1D and 2D, including group velocity fields, micro-support estimates, symplectic vector fields, symbol amplitude/phase, cotangent fiber structure, characteristic sets, wavefront sets, and Hamiltonian flows.
Parameters
pseudo_op:PseudoDifferentialOperator- The pseudo-differential operator whose symbol is to be analyzed interactively.
xlim,ylim:tupleoffloat- Spatial domain limits along x and y axes respectively.
xi_range,eta_range:tuple- Frequency domain limits along ξ and η axes respectively.
density:int- Number of points per axis used to construct the evaluation grid. Controls resolution.
Notes
- In 1D mode, sliders control the fixed frequency (ξ₀) and spatial position (x₀).
- In 2D mode, additional sliders control the second frequency component (η₀) and second spatial coordinate (y₀).
- Visualization updates dynamically as parameters are adjusted via sliders or dropdown menus.
- Supported visualization modes: 'Group Velocity Field' : ∇ξ p(x,ξ) or ∇ p(x,y,ξ,η) 'Micro-Support (1/|p|)' : Reciprocal of symbol magnitude 'Symplectic Vector Field' : (∇_ξ p, -∇_x p) or similar in 2D 'Symbol Amplitude' : |p(x,ξ)| or |p(x,y,ξ,η)| 'Symbol Phase' : arg(p(x,ξ)) or similar in 2D 'Cotangent Fiber' : Structure of symbol over frequency space at fixed x 'Characteristic Set' : Zero set approximation {p ≈ 0} 'Characteristic Gradient' : |∇p(x, ξ)| or |∇p(x₀, y₀, ξ, η)| 'Wavefront Set' : High-frequency singularities detected via symbol interaction 'Hamiltonian Flow' : Trajectories generated by the Hamiltonian vector field
Raises
NotImplementedError- If the spatial dimension is not 1D or 2D.
Prints
Interactive matplotlib figures with dynamic updates based on widget inputs.
Expand source code
def interactive_symbol_analysis(pseudo_op, xlim=(-2, 2), ylim=(-2, 2), xi_range=(0.1, 5), eta_range=(-5, 5), density=100): """ Launch an interactive dashboard for symbol exploration using ipywidgets. This function provides a user-friendly interface to visualize various aspects of the pseudo-differential operator's symbol. It supports multiple visualization modes in both 1D and 2D, including group velocity fields, micro-support estimates, symplectic vector fields, symbol amplitude/phase, cotangent fiber structure, characteristic sets, wavefront sets, and Hamiltonian flows. Parameters ---------- pseudo_op : PseudoDifferentialOperator The pseudo-differential operator whose symbol is to be analyzed interactively. xlim, ylim : tuple of float Spatial domain limits along x and y axes respectively. xi_range, eta_range : tuple Frequency domain limits along ξ and η axes respectively. density : int Number of points per axis used to construct the evaluation grid. Controls resolution. Notes ----- - In 1D mode, sliders control the fixed frequency (ξ₀) and spatial position (x₀). - In 2D mode, additional sliders control the second frequency component (η₀) and second spatial coordinate (y₀). - Visualization updates dynamically as parameters are adjusted via sliders or dropdown menus. - Supported visualization modes: 'Group Velocity Field' : ∇_ξ p(x,ξ) or ∇_{ξ,η} p(x,y,ξ,η) 'Micro-Support (1/|p|)' : Reciprocal of symbol magnitude 'Symplectic Vector Field' : (∇_ξ p, -∇_x p) or similar in 2D 'Symbol Amplitude' : |p(x,ξ)| or |p(x,y,ξ,η)| 'Symbol Phase' : arg(p(x,ξ)) or similar in 2D 'Cotangent Fiber' : Structure of symbol over frequency space at fixed x 'Characteristic Set' : Zero set approximation {p ≈ 0} 'Characteristic Gradient' : |∇p(x, ξ)| or |∇p(x₀, y₀, ξ, η)| 'Wavefront Set' : High-frequency singularities detected via symbol interaction 'Hamiltonian Flow' : Trajectories generated by the Hamiltonian vector field Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. Prints ------ Interactive matplotlib figures with dynamic updates based on widget inputs. """ dim = pseudo_op.dim expr = pseudo_op.expr vars_x = pseudo_op.vars_x mode_selector = Dropdown( options=[ 'Group Velocity Field', 'Micro-Support (1/|p|)', 'Symplectic Vector Field', 'Symbol Amplitude', 'Symbol Phase', 'Cotangent Fiber', 'Characteristic Set', 'Characteristic Gradient', 'Wavefront Set', 'Hamiltonian Flow', ], value='Group Velocity Field', description='Mode:' ) x_vals = np.linspace(*xlim, density) if dim == 2: y_vals = np.linspace(*ylim, density) if dim == 1: x, = vars_x xi = symbols('xi', real=True) grad_func = lambdify((x, xi), diff(expr, xi), 'numpy') symplectic_func = lambdify((x, xi), [diff(expr, xi), -diff(expr, x)], 'numpy') symbol_func = lambdify((x, xi), expr, 'numpy') def plot_1d(mode, xi0, x0): X = x_vals[:, None] if mode == 'Group Velocity Field': V = grad_func(X, xi0) plt.quiver(X, V, np.ones_like(V), V, scale=10, width=0.004) plt.title(f'Group Velocity Field at ξ={xi0:.2f}') elif mode == 'Micro-Support (1/|p|)': Z = 1 / (np.abs(symbol_func(X, xi0)) + 1e-10) plt.plot(x_vals, Z) plt.title(f'Micro-Support (1/|p|) at ξ={xi0:.2f}') elif mode == 'Symplectic Vector Field': U, V = symplectic_func(X, xi0) plt.quiver(X, V, U, V, scale=10, width=0.004) plt.title(f'Symplectic Field at ξ={xi0:.2f}') elif mode == 'Symbol Amplitude': Z = np.abs(symbol_func(X, xi0)) plt.plot(x_vals, Z) plt.title(f'Symbol Amplitude |p(x,ξ)| at ξ={xi0:.2f}') elif mode == 'Symbol Phase': Z = np.angle(symbol_func(X, xi0)) plt.plot(x_vals, Z) plt.title(f'Symbol Phase arg(p(x,ξ)) at ξ={xi0:.2f}') elif mode == 'Cotangent Fiber': pseudo_op.visualize_fiber(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Characteristic Set': pseudo_op.visualize_characteristic_set(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Characteristic Gradient': pseudo_op.visualize_characteristic_gradient(x_vals, np.linspace(*xi_range, density), x0=x0) elif mode == 'Wavefront Set': pseudo_op.visualize_wavefront(x_vals, np.linspace(*xi_range, density), xi0=xi0) elif mode == 'Hamiltonian Flow': pseudo_op.plot_hamiltonian_flow(x0=x0, xi0=xi0) interact(plot_1d, mode=mode_selector, xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'), x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀')) elif dim == 2: x, y = vars_x xi, eta = symbols('xi eta', real=True) grad_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy') symplectic_func = lambdify((x, y, xi, eta), [diff(expr, xi), diff(expr, eta)], 'numpy') symbol_func = lambdify((x, y, xi, eta), expr, 'numpy') def plot_2d(mode, xi0, eta0, x0, y0): X, Y = np.meshgrid(x_vals, y_vals, indexing='ij') if mode == 'Group Velocity Field': U, V = grad_func(X, Y, xi0, eta0) plt.quiver(X, Y, U, V, scale=10, width=0.004) plt.title(f'Group Velocity Field at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Micro-Support (1/|p|)': Z = 1 / (np.abs(symbol_func(X, Y, xi0, eta0)) + 1e-10) plt.pcolormesh(X, Y, Z, shading='auto', cmap='inferno') plt.colorbar(label='1/|p|') plt.title(f'Micro-Support at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symplectic Vector Field': U, V = symplectic_func(X, Y, xi0, eta0) plt.quiver(X, Y, U, V, scale=10, width=0.004) plt.title(f'Symplectic Field at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symbol Amplitude': Z = np.abs(symbol_func(X, Y, xi0, eta0)) plt.pcolormesh(X, Y, Z, shading='auto') plt.colorbar(label='|p(x,y,ξ,η)|') plt.title(f'Symbol Amplitude at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Symbol Phase': Z = np.angle(symbol_func(X, Y, xi0, eta0)) plt.pcolormesh(X, Y, Z, shading='auto', cmap='twilight') plt.colorbar(label='arg(p)') plt.title(f'Symbol Phase at ξ={xi0:.2f}, η={eta0:.2f}') elif mode == 'Cotangent Fiber': pseudo_op.visualize_fiber(np.linspace(*xi_range, density), np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Characteristic Set': pseudo_op.visualize_characteristic_set(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Characteristic Gradient': pseudo_op.visualize_characteristic_gradient(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), x0=x0, y0=y0) elif mode == 'Wavefront Set': pseudo_op.visualize_wavefront(x_grid=x_vals, xi_grid=np.linspace(*xi_range, density), y_grid=y_vals, eta_grid=np.linspace(*eta_range, density), xi0=xi0, eta0=eta0) elif mode == 'Hamiltonian Flow': pseudo_op.plot_hamiltonian_flow(x0=x0, y0=y0, xi0=xi0, eta0=eta0) interact(plot_2d, mode=mode_selector, xi0=FloatSlider(min=xi_range[0], max=xi_range[1], step=0.1, value=1.0, description='ξ₀'), eta0=FloatSlider(min=eta_range[0], max=eta_range[1], step=0.1, value=1.0, description='η₀'), x0=FloatSlider(min=xlim[0], max=xlim[1], step=0.1, value=0.0, description='x₀'), y0=FloatSlider(min=ylim[0], max=ylim[1], step=0.1, value=0.0, description='y₀')) def is_elliptic_numerically(self, x_grid, xi_grid, threshold=1e-08)-
Check if the pseudo-differential symbol p(x, ξ) is elliptic over a given grid.
A symbol is considered elliptic if its magnitude |p(x, ξ)| remains bounded away from zero across all points in the spatial-frequency domain. This method evaluates the symbol on a grid of spatial and frequency coordinates and checks whether its minimum absolute value exceeds a specified threshold.
Resampling is applied to large grids to prevent excessive memory usage, particularly in 2D.
Parameters
x_grid:ndarray- Spatial grid: either a 1D array (x) or a tuple of two 1D arrays (x, y).
xi_grid:ndarray- Frequency grid: either a 1D array (ξ) or a tuple of two 1D arrays (ξ, η).
threshold:float, optional- Minimum acceptable value for |p(x, ξ)|. If the smallest evaluated symbol value falls below this, the symbol is not considered elliptic.
Returns
bool- True if the symbol is elliptic on the resampled grid, False otherwise.
Expand source code
def is_elliptic_numerically(self, x_grid, xi_grid, threshold=1e-8): """ Check if the pseudo-differential symbol p(x, ξ) is elliptic over a given grid. A symbol is considered elliptic if its magnitude |p(x, ξ)| remains bounded away from zero across all points in the spatial-frequency domain. This method evaluates the symbol on a grid of spatial and frequency coordinates and checks whether its minimum absolute value exceeds a specified threshold. Resampling is applied to large grids to prevent excessive memory usage, particularly in 2D. Parameters ---------- x_grid : ndarray Spatial grid: either a 1D array (x) or a tuple of two 1D arrays (x, y). xi_grid : ndarray Frequency grid: either a 1D array (ξ) or a tuple of two 1D arrays (ξ, η). threshold : float, optional Minimum acceptable value for |p(x, ξ)|. If the smallest evaluated symbol value falls below this, the symbol is not considered elliptic. Returns ------- bool True if the symbol is elliptic on the resampled grid, False otherwise. """ RESAMPLE_SIZE = 32 # Reduced size to prevent memory explosion if self.dim == 1: x_vals = x_grid xi_vals = xi_grid # Resampling if necessary if len(x_vals) > RESAMPLE_SIZE: x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE) if len(xi_vals) > RESAMPLE_SIZE: xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') symbol_vals = self.p_func(X, XI) elif self.dim == 2: x_vals, y_vals = x_grid xi_vals, eta_vals = xi_grid # Spatial resampling if len(x_vals) > RESAMPLE_SIZE: x_vals = np.linspace(x_vals.min(), x_vals.max(), RESAMPLE_SIZE) if len(y_vals) > RESAMPLE_SIZE: y_vals = np.linspace(y_vals.min(), y_vals.max(), RESAMPLE_SIZE) # Frequency resampling if len(xi_vals) > RESAMPLE_SIZE: xi_vals = np.linspace(xi_vals.min(), xi_vals.max(), RESAMPLE_SIZE) if len(eta_vals) > RESAMPLE_SIZE: eta_vals = np.linspace(eta_vals.min(), eta_vals.max(), RESAMPLE_SIZE) X, Y, XI, ETA = np.meshgrid(x_vals, y_vals, xi_vals, eta_vals, indexing='ij') symbol_vals = self.p_func(X, Y, XI, ETA) min_abs_val = np.min(np.abs(symbol_vals)) return min_abs_val > threshold def is_homogeneous(self, tol=1e-10)-
Check whether the symbol is homogeneous in the frequency variables.
Returns
(bool, Rational or float or None) Tuple (is_homogeneous, degree) where: - is_homogeneous: True if the symbol satisfies p(λξ, λη) = λ^m * p(ξ, η) - degree: the detected degree m if homogeneous, or None
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def is_homogeneous(self, tol=1e-10): """ Check whether the symbol is homogeneous in the frequency variables. Returns ------- (bool, Rational or float or None) Tuple (is_homogeneous, degree) where: - is_homogeneous: True if the symbol satisfies p(λξ, λη) = λ^m * p(ξ, η) - degree: the detected degree m if homogeneous, or None """ from sympy import symbols, simplify, expand, Eq from sympy.abc import l if self.dim == 1: xi = symbols('xi', real=True, positive=True) l = symbols('l', real=True, positive=True) p = self.expr p_scaled = p.subs(xi, l * xi) ratio = simplify(p_scaled / p) if ratio.has(xi): return False, None try: deg = simplify(ratio).as_base_exp()[1] return True, deg except Exception: return False, None elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) l = symbols('l', real=True, positive=True) p = self.expr p_scaled = p.subs({xi: l * xi, eta: l * eta}) ratio = simplify(p_scaled / p) # If ratio == l**m with no (xi, eta) left, it's homogeneous if ratio.has(xi, eta): return False, None try: base, exp = ratio.as_base_exp() if base == l: return True, exp except Exception: pass return False, None def is_self_adjoint(self, tol=1e-10)-
Check whether the pseudo-differential operator is formally self-adjoint (Hermitian).
A self-adjoint operator satisfies P = P, where P is the formal adjoint of P. This property is essential for ensuring real-valued eigenvalues and stable evolution in quantum mechanics and symmetric wave propagation.
Parameters
tol:float- Tolerance for symbolic comparison between P and P*. Small numerical differences below this threshold are considered equal.
Returns
bool- True if the symbol p(x, ξ) equals its formal adjoint p*(x, ξ) within the given tolerance, indicating that the operator is self-adjoint.
Notes: - The formal adjoint is computed via conjugation and asymptotic expansion at infinity in ξ. - Symbolic simplification is used to verify equality, ensuring robustness against superficial expression differences.
Expand source code
def is_self_adjoint(self, tol=1e-10): """ Check whether the pseudo-differential operator is formally self-adjoint (Hermitian). A self-adjoint operator satisfies P = P*, where P* is the formal adjoint of P. This property is essential for ensuring real-valued eigenvalues and stable evolution in quantum mechanics and symmetric wave propagation. Parameters ---------- tol : float Tolerance for symbolic comparison between P and P*. Small numerical differences below this threshold are considered equal. Returns ------- bool True if the symbol p(x, ξ) equals its formal adjoint p*(x, ξ) within the given tolerance, indicating that the operator is self-adjoint. Notes: - The formal adjoint is computed via conjugation and asymptotic expansion at infinity in ξ. - Symbolic simplification is used to verify equality, ensuring robustness against superficial expression differences. """ p = self.expr p_star = self.formal_adjoint() return simplify(p - p_star).equals(0) def left_inverse_asymptotic(self, order=1)-
Construct a formal left inverse L such that the composition L ∘ P equals the identity operator up to terms of order ξ^{-order}. This expansion is performed asymptotically at infinity in the frequency variable(s).
The left inverse is built iteratively using symbolic differentiation and the method of asymptotic expansions for pseudo-differential operators. It ensures that:
L(P(x,ξ),x,D) ∘ P(x,D) = Id + smoothing operator of order -orderParameters
order:int, optional- Maximum number of terms in the asymptotic expansion (default is 1). Higher values yield more accurate inverses at the cost of increased computational complexity.
Returns
sympy.Expr- Symbolic expression representing the principal symbol of the formal left inverse operator L(x,ξ). This expression depends on spatial variables and frequencies, and includes correction terms up to the specified order.
Notes
- In 1D: Uses recursive application of the Leibniz formula for symbols.
- In 2D: Generalizes to multi-indices for mixed derivatives in (x,y) and (ξ,η).
- Each term involves combinations of derivatives of the original symbol p(x,ξ) and previously computed terms of the inverse.
- Coefficients include powers of 1j (i) and factorial normalization for derivative terms.
Expand source code
def left_inverse_asymptotic(self, order=1): """ Construct a formal left inverse L such that the composition L ∘ P equals the identity operator up to terms of order ξ^{-order}. This expansion is performed asymptotically at infinity in the frequency variable(s). The left inverse is built iteratively using symbolic differentiation and the method of asymptotic expansions for pseudo-differential operators. It ensures that: L(P(x,ξ),x,D) ∘ P(x,D) = Id + smoothing operator of order -order Parameters ---------- order : int, optional Maximum number of terms in the asymptotic expansion (default is 1). Higher values yield more accurate inverses at the cost of increased computational complexity. Returns ------- sympy.Expr Symbolic expression representing the principal symbol of the formal left inverse operator L(x,ξ). This expression depends on spatial variables and frequencies, and includes correction terms up to the specified order. Notes ----- - In 1D: Uses recursive application of the Leibniz formula for symbols. - In 2D: Generalizes to multi-indices for mixed derivatives in (x,y) and (ξ,η). - Each term involves combinations of derivatives of the original symbol p(x,ξ) and previously computed terms of the inverse. - Coefficients include powers of 1j (i) and factorial normalization for derivative terms. """ p = self.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) l = 1 / p.subs(xi, xi) L = l for n in range(1, order + 1): term = 0 for k in range(1, n + 1): coeff = (1j)**(-k) / factorial(k) inner = diff(L, xi, k) * diff(p, x, k) term += coeff * inner L = L - term * l elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) l = 1 / p.subs({xi: xi, eta: eta}) L = l for n in range(1, order + 1): term = 0 for k1 in range(n + 1): for k2 in range(n + 1 - k1): if k1 + k2 == 0: continue coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2)) dp = diff(p, x, k1, y, k2) dL = diff(L, xi, k1, eta, k2) term += coeff * dL * dp L = L - term * l return L def plot_hamiltonian_flow(self, x0=0.0, xi0=5.0, y0=0.0, eta0=0.0, tmax=1.0, n_steps=100, show_field=True)-
Integrate and plot the Hamiltonian trajectories of the symbol in phase space.
This method numerically integrates the Hamiltonian vector field derived from the operator's symbol to visualize how singularities propagate under the flow. It supports both 1D and 2D problems.
Parameters
x0,xi0:float- Initial position and frequency (momentum) in 1D.
y0,eta0:float, optional- Initial position and frequency in 2D; defaults to zero.
tmax:float- Final integration time for the ODE solver.
n_steps:int- Number of time steps used in the integration.
Notes
- The Hamiltonian vector field is obtained from the symplectic flow of the symbol.
- If the field is complex-valued, only its real part is used for integration.
- In 1D, the trajectory is plotted in (x, ξ) phase space.
- In 2D, the spatial trajectory (x(t), y(t)) is shown along with instantaneous momentum vectors (ξ(t), η(t)) using a quiver plot.
Raises
NotImplementedError- If the spatial dimension is not 1D or 2D.
Displays
matplotlib plot Phase space trajectory(ies) showing the evolution of position and momentum under the Hamiltonian dynamics.
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def plot_hamiltonian_flow(self, x0=0.0, xi0=5.0, y0=0.0, eta0=0.0, tmax=1.0, n_steps=100, show_field=True): """ Integrate and plot the Hamiltonian trajectories of the symbol in phase space. This method numerically integrates the Hamiltonian vector field derived from the operator's symbol to visualize how singularities propagate under the flow. It supports both 1D and 2D problems. Parameters ---------- x0, xi0 : float Initial position and frequency (momentum) in 1D. y0, eta0 : float, optional Initial position and frequency in 2D; defaults to zero. tmax : float Final integration time for the ODE solver. n_steps : int Number of time steps used in the integration. Notes ----- - The Hamiltonian vector field is obtained from the symplectic flow of the symbol. - If the field is complex-valued, only its real part is used for integration. - In 1D, the trajectory is plotted in (x, ξ) phase space. - In 2D, the spatial trajectory (x(t), y(t)) is shown along with instantaneous momentum vectors (ξ(t), η(t)) using a quiver plot. Raises ------ NotImplementedError If the spatial dimension is not 1D or 2D. Displays -------- matplotlib plot Phase space trajectory(ies) showing the evolution of position and momentum under the Hamiltonian dynamics. """ def make_real(expr): from sympy import re, simplify expr = expr.doit(deep=True) return simplify(re(expr)) H = self.symplectic_flow() if any(im(H[k]) != 0 for k in H): print("⚠️ The Hamiltonian field is complex. Only the real part is used for integration.") if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) dxdt_expr = make_real(H['dx/dt']) dxidt_expr = make_real(H['dxi/dt']) dxdt = lambdify((x, xi), dxdt_expr, 'numpy') dxidt = lambdify((x, xi), dxidt_expr, 'numpy') def hamilton(t, Y): x, xi = Y return [dxdt(x, xi), dxidt(x, xi)] sol = solve_ivp(hamilton, [0, tmax], [x0, xi0], t_eval=np.linspace(0, tmax, n_steps)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_steps: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_steps = n_points x_vals, xi_vals = sol.y plt.plot(x_vals, xi_vals) plt.xlabel("x") plt.ylabel("ξ") plt.title("Hamiltonian Flow in Phase Space (1D)") plt.grid(True) plt.show() elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) dxdt = lambdify((x, y, xi, eta), make_real(H['dx/dt']), 'numpy') dydt = lambdify((x, y, xi, eta), make_real(H['dy/dt']), 'numpy') dxidt = lambdify((x, y, xi, eta), make_real(H['dxi/dt']), 'numpy') detadt = lambdify((x, y, xi, eta), make_real(H['deta/dt']), 'numpy') def hamilton(t, Y): x, y, xi, eta = Y return [ dxdt(x, y, xi, eta), dydt(x, y, xi, eta), dxidt(x, y, xi, eta), detadt(x, y, xi, eta) ] sol = solve_ivp(hamilton, [0, tmax], [x0, y0, xi0, eta0], t_eval=np.linspace(0, tmax, n_steps)) if sol.status != 0: print(f"⚠️ Integration warning: {sol.message}") n_points = sol.y.shape[1] if n_points < n_steps: print(f"⚠️ Only {n_points} frames computed. Adjusting animation.") n_steps = n_points x_vals, y_vals, xi_vals, eta_vals = sol.y plt.plot(x_vals, y_vals, label='Position') plt.quiver(x_vals, y_vals, xi_vals, eta_vals, scale=20, width=0.003, alpha=0.5, color='r') # Vector field of the flow (optional) if show_field: X, Y = np.meshgrid(np.linspace(min(x_vals), max(x_vals), 20), np.linspace(min(y_vals), max(y_vals), 20)) XI, ETA = xi0 * np.ones_like(X), eta0 * np.ones_like(Y) U = dxdt(X, Y, XI, ETA) V = dydt(X, Y, XI, ETA) plt.quiver(X, Y, U, V, color='gray', alpha=0.2, scale=30, width=0.002) plt.xlabel("x") plt.ylabel("y") plt.title("Hamiltonian Flow in Phase Space (2D)") plt.legend() plt.grid(True) plt.axis('equal') plt.show() def plot_symplectic_vector_field(self, xlim=(-2, 2), klim=(-5, 5), density=30)-
Visualize the symplectic vector field (Hamiltonian vector field) associated with the operator's symbol.
The plotted vector field corresponds to (∂_ξ p, -∂_x p), where p(x, ξ) is the principal symbol of the pseudo-differential operator. This field governs the bicharacteristic flow in phase space.
Parameters
xlim:tupleoffloat- Range for spatial variable x, as (x_min, x_max).
klim:tupleoffloat- Range for frequency variable ξ, as (ξ_min, ξ_max).
density:int- Number of grid points per axis for the visualization grid.
Raises
NotImplementedError- If called on a 2D operator (currently only 1D implementation available).
Notes
- Only supports one-dimensional operators.
- Uses symbolic differentiation to compute ∂_ξ p and ∂_x p.
- Numerical evaluation is done via lambdify with NumPy backend.
- Visualization uses matplotlib quiver plot to show vector directions.
Expand source code
def plot_symplectic_vector_field(self, xlim=(-2, 2), klim=(-5, 5), density=30): """ Visualize the symplectic vector field (Hamiltonian vector field) associated with the operator's symbol. The plotted vector field corresponds to (∂_ξ p, -∂_x p), where p(x, ξ) is the principal symbol of the pseudo-differential operator. This field governs the bicharacteristic flow in phase space. Parameters ---------- xlim : tuple of float Range for spatial variable x, as (x_min, x_max). klim : tuple of float Range for frequency variable ξ, as (ξ_min, ξ_max). density : int Number of grid points per axis for the visualization grid. Raises ------ NotImplementedError If called on a 2D operator (currently only 1D implementation available). Notes ----- - Only supports one-dimensional operators. - Uses symbolic differentiation to compute ∂_ξ p and ∂_x p. - Numerical evaluation is done via lambdify with NumPy backend. - Visualization uses matplotlib quiver plot to show vector directions. """ x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') if self.dim != 1: raise NotImplementedError("Only 1D version implemented.") x, = self.vars_x xi = symbols('xi', real=True) H = self.symplectic_flow() dxdt = lambdify((x, xi), simplify(H['dx/dt']), 'numpy') dxidt = lambdify((x, xi), simplify(H['dxi/dt']), 'numpy') U = dxdt(X, XI) V = dxidt(X, XI) plt.quiver(X, XI, U, V, scale=10, width=0.005) plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Symplectic Vector Field (1D)") plt.grid(True) plt.show() def principal_symbol(self, order=1)-
Compute the leading homogeneous component of the pseudo-differential symbol.
This method extracts the principal part of the symbol, which is the dominant term under high-frequency asymptotics (|ξ| → ∞). The expansion is performed in polar coordinates for 2D symbols to maintain rotational symmetry, then converted back to Cartesian form.
Parameters
order:int- Order of the asymptotic expansion in powers of 1/ρ, where ρ = |ξ| in 1D or ρ = sqrt(ξ² + η²) in 2D. Only the leading-order term is returned.
Returns
sympy.Expr- The principal symbol component, homogeneous of degree
m - order, wheremis the original symbol's order.
Notes: - In 1D, uses direct series expansion in ξ. - In 2D, expands in radial variable ρ while preserving angular dependence. - Useful for microlocal analysis and constructing parametrices.
Expand source code
def principal_symbol(self, order=1): """ Compute the leading homogeneous component of the pseudo-differential symbol. This method extracts the principal part of the symbol, which is the dominant term under high-frequency asymptotics (|ξ| → ∞). The expansion is performed in polar coordinates for 2D symbols to maintain rotational symmetry, then converted back to Cartesian form. Parameters ---------- order : int Order of the asymptotic expansion in powers of 1/ρ, where ρ = |ξ| in 1D or ρ = sqrt(ξ² + η²) in 2D. Only the leading-order term is returned. Returns ------- sympy.Expr The principal symbol component, homogeneous of degree `m - order`, where `m` is the original symbol's order. Notes: - In 1D, uses direct series expansion in ξ. - In 2D, expands in radial variable ρ while preserving angular dependence. - Useful for microlocal analysis and constructing parametrices. """ p = self.expr if self.dim == 1: xi = symbols('xi', real=True, positive=True) return simplify(series(p, xi, oo, n=order).removeO()) elif self.dim == 2: xi, eta = symbols('xi eta', real=True, positive=True) # Homogeneous radial expansion: we set (ξ, η) = ρ (cosθ, sinθ) rho, theta = symbols('rho theta', real=True, positive=True) p_rho = p.subs({xi: rho * cos(theta), eta: rho * sin(theta)}) expansion = series(p_rho, rho, oo, n=order).removeO() # Revert back to (ξ, η) expansion_cart = expansion.subs({rho: sqrt(xi**2 + eta**2), cos(theta): xi / sqrt(xi**2 + eta**2), sin(theta): eta / sqrt(xi**2 + eta**2)}) return simplify(powdenest(expansion_cart, force=True)) def right_inverse_asymptotic(self, order=1)-
Construct a formal right inverse R of the pseudo-differential operator P such that the composition P ∘ R equals the identity plus a smoothing operator of order -order.
This method computes an asymptotic expansion for the right inverse using recursive corrections based on derivatives of the symbol p(x, ξ) and lower-order terms of R.
Parameters
order:int- Number of terms to include in the asymptotic expansion. Higher values improve approximation at the cost of complexity and computational effort.
Returns
sympy.Expr- The symbolic expression representing the formal right inverse R(x, ξ), which satisfies: P ∘ R = Id + O(⟨ξ⟩^{-order}), where ⟨ξ⟩ = (1 + |ξ|²)^{1/2}.
Notes
- In 1D: The recursion involves spatial derivatives of R and derivatives of p with respect to ξ.
- In 2D: The multi-index generalization is used with mixed derivatives in ξ and η.
- The construction relies on the non-vanishing of the principal symbol p to ensure invertibility.
- Each term in the expansion corresponds to higher-order corrections involving commutators between the operator P and the current approximation of R.
Expand source code
def right_inverse_asymptotic(self, order=1): """ Construct a formal right inverse R of the pseudo-differential operator P such that the composition P ∘ R equals the identity plus a smoothing operator of order -order. This method computes an asymptotic expansion for the right inverse using recursive corrections based on derivatives of the symbol p(x, ξ) and lower-order terms of R. Parameters ---------- order : int Number of terms to include in the asymptotic expansion. Higher values improve approximation at the cost of complexity and computational effort. Returns ------- sympy.Expr The symbolic expression representing the formal right inverse R(x, ξ), which satisfies: P ∘ R = Id + O(⟨ξ⟩^{-order}), where ⟨ξ⟩ = (1 + |ξ|²)^{1/2}. Notes ----- - In 1D: The recursion involves spatial derivatives of R and derivatives of p with respect to ξ. - In 2D: The multi-index generalization is used with mixed derivatives in ξ and η. - The construction relies on the non-vanishing of the principal symbol p to ensure invertibility. - Each term in the expansion corresponds to higher-order corrections involving commutators between the operator P and the current approximation of R. """ p = self.expr if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True) r = 1 / p.subs(xi, xi) # r0 R = r for n in range(1, order + 1): term = 0 for k in range(1, n + 1): coeff = (1j)**(-k) / factorial(k) inner = diff(p, xi, k) * diff(R, x, k) term += coeff * inner R = R - r * term elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) r = 1 / p.subs({xi: xi, eta: eta}) R = r for n in range(1, order + 1): term = 0 for k1 in range(n + 1): for k2 in range(n + 1 - k1): if k1 + k2 == 0: continue coeff = (1j)**(-(k1 + k2)) / (factorial(k1) * factorial(k2)) dp = diff(p, xi, k1, eta, k2) dR = diff(R, x, k1, y, k2) term += coeff * dp * dR R = R - r * term return R def simulate_evolution(self, x_grid, t_grid, y_grid=None, initial_condition=None, initial_velocity=None, solver_params=None, component='real')-
Simulate and animate the time evolution of an initial condition under a pseudo-differential operator.
This method solves the PDE governed by the action of a pseudo-differential operator
p(x, D), either as a first-order or second-order time evolution, and generates an animation of the resulting wave propagation. The order of the PDE depends on whether an initial velocity is provided.Parameters
x_grid:numpy.ndarray- Spatial grid in the x-direction.
t_grid:numpy.ndarray- Temporal grid for the simulation.
y_grid:numpy.ndarray, optional- Spatial grid in the y-direction (used for 2D problems).
initial_condition:callable, optional- Function specifying the initial condition u₀(x) or u₀(x, y).
initial_velocity:callable, optional- Function specifying the initial velocity ∂ₜu₀(x) or ∂ₜu₀(x, y). If provided, a second-order PDE is solved.
solver_params:dict, optional- Additional keyword arguments passed to the PDE solver.
component:{'real', 'imag', 'abs', 'angle'}- Specifies which component of the solution to animate.
Returns
matplotlib.animation.FuncAnimation- Animation object showing the evolution of the solution over time.
Notes
- Solves the PDE:
- First-order: ∂ₜu = -i ⋅ p(x, D) u
- Second-order: ∂²ₜu = -p(x, D)² u
- Supports 1D and 2D simulations depending on the presence of
y_grid. - The solution is animated using the specified component view.
- Useful for visualizing wave propagation and dispersive effects driven by pseudo-differential operators.
Expand source code
def simulate_evolution(self, x_grid, t_grid, y_grid=None, initial_condition=None, initial_velocity=None, solver_params=None, component='real'): """ Simulate and animate the time evolution of an initial condition under a pseudo-differential operator. This method solves the PDE governed by the action of a pseudo-differential operator `p(x, D)`, either as a first-order or second-order time evolution, and generates an animation of the resulting wave propagation. The order of the PDE depends on whether an initial velocity is provided. Parameters ---------- x_grid : numpy.ndarray Spatial grid in the x-direction. t_grid : numpy.ndarray Temporal grid for the simulation. y_grid : numpy.ndarray, optional Spatial grid in the y-direction (used for 2D problems). initial_condition : callable, optional Function specifying the initial condition u₀(x) or u₀(x, y). initial_velocity : callable, optional Function specifying the initial velocity ∂ₜu₀(x) or ∂ₜu₀(x, y). If provided, a second-order PDE is solved. solver_params : dict, optional Additional keyword arguments passed to the PDE solver. component : {'real', 'imag', 'abs', 'angle'} Specifies which component of the solution to animate. Returns ------- matplotlib.animation.FuncAnimation Animation object showing the evolution of the solution over time. Notes ----- - Solves the PDE: - First-order: ∂ₜu = -i ⋅ p(x, D) u - Second-order: ∂²ₜu = -p(x, D)² u - Supports 1D and 2D simulations depending on the presence of `y_grid`. - The solution is animated using the specified component view. - Useful for visualizing wave propagation and dispersive effects driven by pseudo-differential operators. """ if solver_params is None: solver_params = {} # --- 1. Symbolic variables --- t = symbols('t', real=True) u_sym = Function('u') is_second_order = initial_velocity is not None if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) u = u_sym(t, x) if is_second_order: eq = Eq(diff(u, t, 2), psiOp(self.expr, u)) else: eq = Eq(diff(u, t), psiOp(self.expr, u)) elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) u = u_sym(t, x, y) if is_second_order: eq = Eq(diff(u, t, 2), psiOp(self.expr, u)) else: eq = Eq(diff(u, t), psiOp(self.expr, u)) else: raise NotImplementedError("Only 1D and 2D are supported.") # --- 2. Create the solver --- solver = PDESolver(eq) params = { 'Lx': x_grid.max() - x_grid.min(), 'Nx': len(x_grid), 'Lt': t_grid.max() - t_grid.min(), 'Nt': len(t_grid), 'boundary_condition': 'periodic', 'n_frames': min(100, len(t_grid)) } if self.dim == 2: params['Ly'] = y_grid.max() - y_grid.min() params['Ny'] = len(y_grid) params.update(solver_params) # --- 3. Initial condition --- if initial_condition is None: raise ValueError("initial_condition is None. Please provide a function u₀(x) or u₀(x, y) as the initial condition.") params['initial_condition'] = initial_condition if is_second_order: params['initial_velocity'] = initial_velocity # --- 4. Solving --- print("⚙️ Solving the evolution equation (order {} in time)...".format(2 if is_second_order else 1)) solver.setup(**params) solver.solve() print("✅ Solving completed.") # --- 5. Animation --- print("🎞️ Creating the animation...") ani = solver.animate(component=component) return ani def symbol_order(self, max_order=10, tol=0.001)-
Estimate the homogeneity order of the pseudo-differential symbol in high-frequency asymptotics.
This method attempts to determine the leading-order behavior of the symbol p(x, ξ) or p(x, y, ξ, η) as |ξ| → ∞ (in 1D) or |(ξ, η)| → ∞ (in 2D). The returned value represents the asymptotic growth or decay rate, which is essential for understanding the regularity and mapping properties of the corresponding operator.
The function uses symbolic preprocessing to ensure proper factorization of frequency variables, especially in sqrt and power expressions, to avoid erroneous order detection (e.g., due to hidden scaling).
Parameters
max_order:int, optional- Maximum number of terms to consider in the series expansion. Default is 10.
tol:float, optional- Tolerance threshold for evaluating the coefficient magnitude. If the coefficient is too small, the detected order may be discarded. Default is 1e-3.
Returns
floatorNone-
- If the symbol is homogeneous, returns its exact homogeneity degree as a float.
- Otherwise, estimates the dominant asymptotic order from leading terms in the expansion.
- Returns None if no valid order could be determined.
Notes
-
In 1D: Two strategies are used: 1. Expand directly in xi at infinity. 2. Substitute xi = 1/z and expand around z = 0.
-
In 2D:
- Transform the symbol into polar coordinates: (xi, eta) = rho*(cos(theta), sin(theta)).
- Expand in rho at infinity, then extract the leading term's power.
- An alternative substitution using 1/z is also tried if the first method fails.
-
Preprocessing steps:
- Sqrt expressions involving frequencies are rewritten to isolate the leading variable.
- Power expressions are factored explicitly to ensure correct symbolic scaling.
-
If the symbol is not homogeneous, a warning is issued, and the result should be interpreted with care.
-
For non-homogeneous symbols, only the principal asymptotic term is considered.
Raises
NotImplementedError- If the spatial dimension is neither 1 nor 2.
Expand source code
def symbol_order(self, max_order=10, tol=1e-3): """ Estimate the homogeneity order of the pseudo-differential symbol in high-frequency asymptotics. This method attempts to determine the leading-order behavior of the symbol p(x, ξ) or p(x, y, ξ, η) as |ξ| → ∞ (in 1D) or |(ξ, η)| → ∞ (in 2D). The returned value represents the asymptotic growth or decay rate, which is essential for understanding the regularity and mapping properties of the corresponding operator. The function uses symbolic preprocessing to ensure proper factorization of frequency variables, especially in sqrt and power expressions, to avoid erroneous order detection (e.g., due to hidden scaling). Parameters ---------- max_order : int, optional Maximum number of terms to consider in the series expansion. Default is 10. tol : float, optional Tolerance threshold for evaluating the coefficient magnitude. If the coefficient is too small, the detected order may be discarded. Default is 1e-3. Returns ------- float or None - If the symbol is homogeneous, returns its exact homogeneity degree as a float. - Otherwise, estimates the dominant asymptotic order from leading terms in the expansion. - Returns None if no valid order could be determined. Notes ----- - In 1D: Two strategies are used: 1. Expand directly in xi at infinity. 2. Substitute xi = 1/z and expand around z = 0. - In 2D: - Transform the symbol into polar coordinates: (xi, eta) = rho*(cos(theta), sin(theta)). - Expand in rho at infinity, then extract the leading term's power. - An alternative substitution using 1/z is also tried if the first method fails. - Preprocessing steps: - Sqrt expressions involving frequencies are rewritten to isolate the leading variable. - Power expressions are factored explicitly to ensure correct symbolic scaling. - If the symbol is not homogeneous, a warning is issued, and the result should be interpreted with care. - For non-homogeneous symbols, only the principal asymptotic term is considered. Raises ------ NotImplementedError If the spatial dimension is neither 1 nor 2. """ from sympy import ( symbols, series, simplify, sqrt, cos, sin, oo, powdenest, radsimp, expand, expand_power_base ) def preprocess_sqrt(expr, freq): return expr.replace( lambda e: e.func == sqrt and freq in e.free_symbols, lambda e: freq * sqrt(1 + (e.args[0] - freq**2) / freq**2) ) def preprocess_power(expr, freq): return expr.replace( lambda e: e.is_Pow and freq in e.free_symbols, lambda e: freq**e.exp * (1 + e.base / freq**e.base.as_powers_dict().get(freq, 0))**e.exp ) def validate_order(power, coeff, vars_x, tol): if power is None: return None if any(v in coeff.free_symbols for v in vars_x): print("⚠️ Coefficient depends on spatial variables; ignoring") return None try: coeff_val = abs(float(coeff.evalf())) if coeff_val < tol: print(f"⚠️ Coefficient too small ({coeff_val:.2e} < {tol})") return None except Exception as e: print(f"⚠️ Coefficient evaluation failed: {e}") return None return int(power) if power == int(power) else float(power) # Homogeneity check is_homog, degree = self.is_homogeneous() if is_homog: return float(degree) else: print("⚠️ The symbol is not homogeneous. The asymptotic order is not well defined.") if self.dim == 1: x = self.vars_x[0] xi = symbols('xi', real=True, positive=True) try: print("1D symbol_order - method 1") expr = preprocess_sqrt(self.expr, xi) s = series(expr, xi, oo, n=max_order).removeO() lead = simplify(powdenest(s.as_leading_term(xi), force=True)) power = lead.as_powers_dict().get(xi, None) coeff = lead / xi**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x], tol) if order is not None: return order except Exception: pass try: print("1D symbol_order - method 2") z = symbols('z', real=True, positive=True) expr_z = preprocess_sqrt(self.expr.subs(xi, 1/z), 1/z) s = series(expr_z, z, 0, n=max_order).removeO() lead = simplify(powdenest(s.as_leading_term(z), force=True)) power = lead.as_powers_dict().get(z, None) coeff = lead / z**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x], tol) if order is not None: return -order except Exception as e: print(f"⚠️ fallback z failed: {e}") return None elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True, positive=True) rho, theta = symbols('rho theta', real=True, positive=True) try: print("2D symbol_order - method 1") p_rho = self.expr.subs({xi: rho * cos(theta), eta: rho * sin(theta)}) p_rho = preprocess_power(preprocess_sqrt(p_rho, rho), rho) s = series(simplify(p_rho), rho, oo, n=max_order).removeO() lead = radsimp(simplify(powdenest(s.as_leading_term(rho), force=True))) power = lead.as_powers_dict().get(rho, None) coeff = lead / rho**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x, y], tol) if order is not None: return order except Exception as e: print(f"⚠️ polar expansion failed: {e}") try: print("2D symbol_order - method 2") z = symbols('z', real=True, positive=True) xi_eta = {xi: (1/z) * cos(theta), eta: (1/z) * sin(theta)} p_rho = preprocess_sqrt(self.expr.subs(xi_eta), 1/z) s = series(simplify(p_rho), z, 0, n=max_order).removeO() lead = radsimp(simplify(powdenest(s.as_leading_term(z), force=True))) power = lead.as_powers_dict().get(z, None) coeff = lead / z**power if power is not None else 0 print("lead =", lead) print("power =", power) print("coeff =", coeff) order = validate_order(power, coeff, [x, y], tol) if order is not None: return -order except Exception as e: print(f"⚠️ fallback z (2D) failed: {e}") return None else: raise NotImplementedError("Only 1D and 2D supported.") def symplectic_flow(self)-
Compute the Hamiltonian vector field associated with the principal symbol.
This method derives the canonical equations of motion for the phase space variables (x, ξ) in 1D or (x, y, ξ, η) in 2D, based on the Hamiltonian formalism. These describe how position and frequency variables evolve under the flow generated by the symbol.
Returns
dict- A dictionary containing the components of the Hamiltonian vector field: - In 1D: keys are 'dx/dt' and 'dxi/dt', corresponding to dx/dt = ∂p/∂ξ and dξ/dt = -∂p/∂x. - In 2D: keys are 'dx/dt', 'dy/dt', 'dxi/dt', and 'deta/dt', with similar definitions: dx/dt = ∂p/∂ξ, dy/dt = ∂p/∂η, dξ/dt = -∂p/∂x, dη/dt = -∂p/∂y.
Notes
- The Hamiltonian here is the principal symbol p(x, ξ) itself.
- This flow preserves the symplectic structure of phase space.
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def symplectic_flow(self): """ Compute the Hamiltonian vector field associated with the principal symbol. This method derives the canonical equations of motion for the phase space variables (x, ξ) in 1D or (x, y, ξ, η) in 2D, based on the Hamiltonian formalism. These describe how position and frequency variables evolve under the flow generated by the symbol. Returns ------- dict A dictionary containing the components of the Hamiltonian vector field: - In 1D: keys are 'dx/dt' and 'dxi/dt', corresponding to dx/dt = ∂p/∂ξ and dξ/dt = -∂p/∂x. - In 2D: keys are 'dx/dt', 'dy/dt', 'dxi/dt', and 'deta/dt', with similar definitions: dx/dt = ∂p/∂ξ, dy/dt = ∂p/∂η, dξ/dt = -∂p/∂x, dη/dt = -∂p/∂y. Notes ----- - The Hamiltonian here is the principal symbol p(x, ξ) itself. - This flow preserves the symplectic structure of phase space. """ if self.dim == 1: x, = self.vars_x xi = symbols('xi', real=True) print("x = ", x) print("xi = ", xi) return { 'dx/dt': diff(self.symbol, xi), 'dxi/dt': -diff(self.symbol, x) } elif self.dim == 2: x, y = self.vars_x xi, eta = symbols('xi eta', real=True) return { 'dx/dt': diff(self.symbol, xi), 'dy/dt': diff(self.symbol, eta), 'dxi/dt': -diff(self.symbol, x), 'deta/dt': -diff(self.symbol, y) } def visualize_characteristic_gradient(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0)-
Visualize the norm of the gradient of the symbol in phase space.
This method computes the magnitude of the gradient |∇p| of a pseudo-differential symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. The resulting colormap reveals regions where the symbol varies rapidly or remains nearly stationary, which is particularly useful for analyzing characteristic sets.
Parameters
x_grid:numpy.ndarray- 1D array of spatial coordinates for the x-direction.
xi_grid:numpy.ndarray- 1D array of frequency coordinates (ξ).
y_grid:numpy.ndarray, optional- 1D array of spatial coordinates for the y-direction (used in 2D mode). Default is None.
eta_grid:numpy.ndarray, optional- 1D array of frequency coordinates (η) for the 2D case. Default is None.
x0:float, optional- Fixed x-coordinate for evaluating the symbol in 2D. Default is 0.0.
y0:float, optional- Fixed y-coordinate for evaluating the symbol in 2D. Default is 0.0.
Returns
None- Displays a 2D colormap of |∇p| over the relevant phase-space domain.
Notes
- In 1D, the full gradient ∇p = (∂ₓp, ∂ξp) is computed over the (x, ξ) grid.
- In 2D, the gradient ∇p = (∂ξp, ∂ηp) is computed at a fixed spatial point (x₀, y₀) over the (ξ, η) grid.
- Numerical differentiation is performed using
np.gradient. - High values of |∇p| indicate rapid variation of the symbol, while low values typically suggest characteristic regions.
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def visualize_characteristic_gradient(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0): """ Visualize the norm of the gradient of the symbol in phase space. This method computes the magnitude of the gradient |∇p| of a pseudo-differential symbol p(x, ξ) in 1D or p(x, y, ξ, η) in 2D. The resulting colormap reveals regions where the symbol varies rapidly or remains nearly stationary, which is particularly useful for analyzing characteristic sets. Parameters ---------- x_grid : numpy.ndarray 1D array of spatial coordinates for the x-direction. xi_grid : numpy.ndarray 1D array of frequency coordinates (ξ). y_grid : numpy.ndarray, optional 1D array of spatial coordinates for the y-direction (used in 2D mode). Default is None. eta_grid : numpy.ndarray, optional 1D array of frequency coordinates (η) for the 2D case. Default is None. x0 : float, optional Fixed x-coordinate for evaluating the symbol in 2D. Default is 0.0. y0 : float, optional Fixed y-coordinate for evaluating the symbol in 2D. Default is 0.0. Returns ------- None Displays a 2D colormap of |∇p| over the relevant phase-space domain. Notes ----- - In 1D, the full gradient ∇p = (∂ₓp, ∂ξp) is computed over the (x, ξ) grid. - In 2D, the gradient ∇p = (∂ξp, ∂ηp) is computed at a fixed spatial point (x₀, y₀) over the (ξ, η) grid. - Numerical differentiation is performed using `np.gradient`. - High values of |∇p| indicate rapid variation of the symbol, while low values typically suggest characteristic regions. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) grad_x = np.gradient(symbol_vals, axis=0) grad_xi = np.gradient(symbol_vals, axis=1) grad_norm = np.sqrt(grad_x**2 + grad_xi**2) plt.pcolormesh(X, XI, grad_norm, cmap='inferno', shading='auto') plt.colorbar(label='|∇p|') plt.title('Gradient Norm (High Near Zeros)') plt.grid(True) plt.show() elif self.dim == 2: xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij') symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) grad_xi = np.gradient(symbol_vals, axis=0) grad_eta = np.gradient(symbol_vals, axis=1) grad_norm = np.sqrt(grad_xi**2 + grad_eta**2) plt.pcolormesh(xi_grid, eta_grid, grad_norm, cmap='inferno', shading='auto') plt.colorbar(label='|∇p|') plt.title(f'Gradient Norm at x={x0}, y={y0}') plt.grid(True) plt.show() def visualize_characteristic_set(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0, levels=[0.1])-
Visualize the characteristic set of the pseudo-differential symbol, defined as the approximate zero set p(x, ξ) ≈ 0.
In microlocal analysis, the characteristic set is the locus of points in phase space (x, ξ) where the symbol p(x, ξ) vanishes, playing a key role in understanding propagation of singularities and wavefronts.
Parameters
x_grid:ndarray- Spatial grid values (1D array) for plotting in 1D or evaluation point in 2D.
xi_grid:ndarray- Frequency variable grid values (1D array) used to construct the frequency domain.
x0:float, optional- Fixed spatial coordinate in 2D case for evaluating the symbol at a specific x position.
y0:float, optional- Fixed spatial coordinate in 2D case for evaluating the symbol at a specific y position.
Notes
- For 1D, this method plots the contour of |p(x, ξ)| = ε with ε = 1e-5 over the (x, ξ) plane.
- For 2D, it evaluates the symbol at fixed (x₀, y₀) and plots the characteristic set in the (ξ, η) frequency plane.
- This visualization helps identify directions of degeneracy or hypoellipticity of the operator.
Raises
NotImplementedError- If called on a solver with dimensionality other than 1D or 2D.
Displays
A matplotlib contour plot showing either: - The characteristic curve in the (x, ξ) phase plane (1D), - The characteristic surface slice in the (ξ, η) frequency plane at (x₀, y₀) (2D).
Expand source code
def visualize_characteristic_set(self, x_grid, xi_grid, y_grid=None, eta_grid=None, y0=0.0, x0=0.0, levels=[1e-1]): """ Visualize the characteristic set of the pseudo-differential symbol, defined as the approximate zero set p(x, ξ) ≈ 0. In microlocal analysis, the characteristic set is the locus of points in phase space (x, ξ) where the symbol p(x, ξ) vanishes, playing a key role in understanding propagation of singularities and wavefronts. Parameters ---------- x_grid : ndarray Spatial grid values (1D array) for plotting in 1D or evaluation point in 2D. xi_grid : ndarray Frequency variable grid values (1D array) used to construct the frequency domain. x0 : float, optional Fixed spatial coordinate in 2D case for evaluating the symbol at a specific x position. y0 : float, optional Fixed spatial coordinate in 2D case for evaluating the symbol at a specific y position. Notes ----- - For 1D, this method plots the contour of |p(x, ξ)| = ε with ε = 1e-5 over the (x, ξ) plane. - For 2D, it evaluates the symbol at fixed (x₀, y₀) and plots the characteristic set in the (ξ, η) frequency plane. - This visualization helps identify directions of degeneracy or hypoellipticity of the operator. Raises ------ NotImplementedError If called on a solver with dimensionality other than 1D or 2D. Displays ------ A matplotlib contour plot showing either: - The characteristic curve in the (x, ξ) phase plane (1D), - The characteristic surface slice in the (ξ, η) frequency plane at (x₀, y₀) (2D). """ if self.dim == 1: x_grid = np.asarray(x_grid) xi_grid = np.asarray(xi_grid) X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.contour(X, XI, np.abs(symbol_vals), levels=levels, colors='red') plt.xlabel('x') plt.ylabel('ξ') plt.title('Characteristic Set (p(x, ξ) ≈ 0)') plt.grid(True) plt.show() elif self.dim == 2: if eta_grid is None: raise ValueError("eta_grid must be provided for 2D visualization.") xi_grid = np.asarray(xi_grid) eta_grid = np.asarray(eta_grid) xi_grid2, eta_grid2 = np.meshgrid(xi_grid, eta_grid, indexing='ij') symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) plt.contour(xi_grid, eta_grid, np.abs(symbol_vals), levels=levels, colors='red') plt.xlabel('ξ') plt.ylabel('η') plt.title(f'Characteristic Set at x={x0}, y={y0}') plt.grid(True) plt.show() else: raise NotImplementedError("Only 1D/2D characteristic sets supported.") def visualize_dynamic_wavefront(self, x_grid, t_grid, y_grid=None, xi0=5.0, eta0=0.0)-
Create an animation of a monochromatic wave evolving under the pseudo-differential operator.
This method visualizes the time evolution of a wavefront governed by the symbol of the pseudo-differential operator. The wave is initialized as a monochromatic solution with fixed spatial frequencies and evolves according to the dispersion relation ω = p(x, ξ), where p is the symbol of the operator and (ξ, η) are spatial frequencies.
Parameters
x_grid:array_like- 1D array representing the spatial grid in the x-direction.
t_grid:array_like- 1D array of time points used to generate the animation frames.
y_grid:array_like, optional- 1D array representing the spatial grid in the y-direction (required for 2D operators).
xi0:float, optional- Initial spatial frequency in the x-direction. Default is 5.0.
eta0:float, optional- Initial spatial frequency in the y-direction (used in 2D). Default is 0.0.
Returns
matplotlib.animation.FuncAnimation- Animation object showing the time evolution of the monochromatic wave.
Notes
- In 1D, visualizes the wave u(x, t) = cos(ξ₀·x − ω·t).
- In 2D, visualizes u(x, y, t) = cos(ξ₀·x + η₀·y − ω·t).
- The frequency ω is computed as the symbol evaluated at the fixed frequency components.
- If the symbol depends on space, it is evaluated at the origin (x=0, y=0).
Raises
ValueError- If
y_gridis not provided while the operator is 2D. NotImplementedError- If the spatial dimension of the operator is neither 1 nor 2.
Expand source code
def visualize_dynamic_wavefront(self, x_grid, t_grid, y_grid=None, xi0=5.0, eta0=0.0): """ Create an animation of a monochromatic wave evolving under the pseudo-differential operator. This method visualizes the time evolution of a wavefront governed by the symbol of the pseudo-differential operator. The wave is initialized as a monochromatic solution with fixed spatial frequencies and evolves according to the dispersion relation ω = p(x, ξ), where p is the symbol of the operator and (ξ, η) are spatial frequencies. Parameters ---------- x_grid : array_like 1D array representing the spatial grid in the x-direction. t_grid : array_like 1D array of time points used to generate the animation frames. y_grid : array_like, optional 1D array representing the spatial grid in the y-direction (required for 2D operators). xi0 : float, optional Initial spatial frequency in the x-direction. Default is 5.0. eta0 : float, optional Initial spatial frequency in the y-direction (used in 2D). Default is 0.0. Returns ------- matplotlib.animation.FuncAnimation Animation object showing the time evolution of the monochromatic wave. Notes ----- - In 1D, visualizes the wave u(x, t) = cos(ξ₀·x − ω·t). - In 2D, visualizes u(x, y, t) = cos(ξ₀·x + η₀·y − ω·t). - The frequency ω is computed as the symbol evaluated at the fixed frequency components. - If the symbol depends on space, it is evaluated at the origin (x=0, y=0). Raises ------ ValueError If `y_grid` is not provided while the operator is 2D. NotImplementedError If the spatial dimension of the operator is neither 1 nor 2. """ if self.dim == 1: # Calculer omega a partir du symbole from sympy import symbols, N x = self.vars_x[0] xi = symbols('xi', real=True) try: omega_symbolic = self.expr.subs({x: 0, xi: xi0}) except: omega_symbolic = self.expr.subs(xi, xi0) omega_val = float(N(omega_symbolic.as_real_imag()[0])) # Preparer les donnees pour l'animation X, T = np.meshgrid(x_grid, t_grid, indexing='ij') # Initialiser la figure et l'axe fig, ax = plt.subplots() ax.set_xlim(x_grid.min(), x_grid.max()) ax.set_ylim(-1.1, 1.1) ax.set_xlabel('x') ax.set_ylabel('u(x, t)') ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t)') ax.grid(True) line, = ax.plot([], [], lw=2) # Pre-calculer les temps T_vals = t_grid def animate(frame): t = T_vals[frame] if frame < len(T_vals) else T_vals[-1] U = np.cos(xi0 * x_grid - omega_val * t) line.set_data(x_grid, U) ax.set_title(f'Dynamic 1D Wavefront u(x, t) = cos({xi0}*x - {omega_val:.2f}*t) at t={t:.2f}') return line, ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=True, repeat=True) plt.show() return ani elif self.dim == 2: # Calculer omega a partir du symbole from sympy import symbols, N x, y = self.vars_x xi, eta = symbols('xi eta', real=True) try: omega_symbolic = self.expr.subs({x: 0, y: 0, xi: xi0, eta: eta0}) except: omega_symbolic = self.expr.subs({xi: xi0, eta: eta0}) omega_val = float(N(omega_symbolic.as_real_imag()[0])) if y_grid is None: raise ValueError("y_grid must be provided for 2D visualization.") # Creer les grilles X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') # Initialiser la figure fig, ax = plt.subplots() ax.set_xlabel('x') ax.set_ylabel('y') # Premiere frame t0 = t_grid[0] if len(t_grid) > 0 else 0.0 U = np.cos(xi0 * X + eta0 * Y - omega_val * t0) extent = [x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()] im = ax.imshow(U, extent=extent, aspect='equal', origin='lower', cmap='seismic', vmin=-1, vmax=1) ax.set_title(f'Dynamic 2D Wavefront at t={t0:.2f}') fig.colorbar(im, ax=ax, label='u(x, y, t)') # Pre-calculer les temps T_vals = t_grid def animate(frame): t = T_vals[frame] if frame < len(T_vals) else T_vals[-1] U = np.cos(xi0 * X + eta0 * Y - omega_val * t) im.set_array(U) ax.set_title(f'Dynamic 2D Wavefront at t={t:.2f}') return [im] ani = animation.FuncAnimation(fig, animate, frames=len(T_vals), interval=50, blit=False, repeat=True) plt.show() return ani def visualize_fiber(self, x_grid, xi_grid, y0=0.0, x0=0.0)-
Plot the cotangent fiber structure at a fixed spatial point (x₀[, y₀]).
This visualization shows how the symbol p(x, ξ) behaves on the cotangent fiber above a fixed spatial point. In microlocal analysis, this provides insight into the frequency content of the operator at that location.
Parameters
x_grid:ndarray- Spatial grid values (1D) for evaluation in 1D case.
xi_grid:ndarray- Frequency grid values (1D) for evaluation in both 1D and 2D cases.
x0:float, optional- Fixed x-coordinate of the base point in space (1D or 2D).
y0:float, optional- Fixed y-coordinate of the base point in space (2D only).
Notes
- In 1D: Displays |p(x, ξ)| over the (x, ξ) phase plane near the fixed point.
- In 2D: Fixes (x₀, y₀) and evaluates p(x₀, y₀, ξ, η), showing the fiber over that point.
- The color map represents the magnitude of the symbol, highlighting regions where it vanishes or becomes singular.
Raises
NotImplementedError- If called in 2D with missing or improperly formatted grids.
Expand source code
def visualize_fiber(self, x_grid, xi_grid, y0=0.0, x0=0.0): """ Plot the cotangent fiber structure at a fixed spatial point (x₀[, y₀]). This visualization shows how the symbol p(x, ξ) behaves on the cotangent fiber above a fixed spatial point. In microlocal analysis, this provides insight into the frequency content of the operator at that location. Parameters ---------- x_grid : ndarray Spatial grid values (1D) for evaluation in 1D case. xi_grid : ndarray Frequency grid values (1D) for evaluation in both 1D and 2D cases. x0 : float, optional Fixed x-coordinate of the base point in space (1D or 2D). y0 : float, optional Fixed y-coordinate of the base point in space (2D only). Notes ----- - In 1D: Displays |p(x, ξ)| over the (x, ξ) phase plane near the fixed point. - In 2D: Fixes (x₀, y₀) and evaluates p(x₀, y₀, ξ, η), showing the fiber over that point. - The color map represents the magnitude of the symbol, highlighting regions where it vanishes or becomes singular. Raises ------ NotImplementedError If called in 2D with missing or improperly formatted grids. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.contourf(X, XI, np.abs(symbol_vals), levels=50, cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('x (position)') plt.ylabel('ξ (frequency)') plt.title('Cotangent Fiber Structure') plt.show() elif self.dim == 2: xi_grid2, eta_grid2 = np.meshgrid(xi_grid, xi_grid) symbol_vals = self.p_func(x0, y0, xi_grid2, eta_grid2) plt.contourf(xi_grid, xi_grid, np.abs(symbol_vals), levels=50, cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('ξ') plt.ylabel('η') plt.title(f'Cotangent Fiber at x={x0}, y={y0}') plt.show() def visualize_micro_support(self, xlim=(-2, 2), klim=(-10, 10), threshold=0.001, density=300)-
Visualize the micro-support of the operator by plotting the inverse of the symbol magnitude 1 / |p(x, ξ)|.
The micro-support provides insight into the singularities of a pseudo-differential operator in phase space (x, ξ). Regions where |p(x, ξ)| is small correspond to large values in 1/|p(x, ξ)|, highlighting areas of significant operator influence or singularity.
Parameters
xlim:tuple- Spatial domain limits (x_min, x_max).
klim:tuple- Frequency domain limits (ξ_min, ξ_max).
threshold:float- Threshold below which |p(x, ξ)| is considered effectively zero; used for numerical stability.
density:int- Number of grid points along each axis for visualization resolution.
Raises
NotImplementedError- If called on a solver with dimension greater than 1 (only 1D visualization is supported).
Notes
- This method evaluates the symbol p(x, ξ) over a grid and plots its reciprocal to emphasize regions where the symbol is near zero.
- A small constant (1e-10) is added to the denominator to avoid division by zero.
- The resulting plot helps identify characteristic sets and wavefront set approximations.
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def visualize_micro_support(self, xlim=(-2, 2), klim=(-10, 10), threshold=1e-3, density=300): """ Visualize the micro-support of the operator by plotting the inverse of the symbol magnitude 1 / |p(x, ξ)|. The micro-support provides insight into the singularities of a pseudo-differential operator in phase space (x, ξ). Regions where |p(x, ξ)| is small correspond to large values in 1/|p(x, ξ)|, highlighting areas of significant operator influence or singularity. Parameters ---------- xlim : tuple Spatial domain limits (x_min, x_max). klim : tuple Frequency domain limits (ξ_min, ξ_max). threshold : float Threshold below which |p(x, ξ)| is considered effectively zero; used for numerical stability. density : int Number of grid points along each axis for visualization resolution. Raises ------ NotImplementedError If called on a solver with dimension greater than 1 (only 1D visualization is supported). Notes ----- - This method evaluates the symbol p(x, ξ) over a grid and plots its reciprocal to emphasize regions where the symbol is near zero. - A small constant (1e-10) is added to the denominator to avoid division by zero. - The resulting plot helps identify characteristic sets and wavefront set approximations. """ if self.dim != 1: raise NotImplementedError("Only 1D micro-support visualization implemented.") x_vals = np.linspace(*xlim, density) xi_vals = np.linspace(*klim, density) X, XI = np.meshgrid(x_vals, xi_vals, indexing='ij') Z = np.abs(self.p_func(X, XI)) plt.contourf(X, XI, 1 / (Z + 1e-10), levels=100, cmap='inferno') plt.colorbar(label=r'$1/|p(x,\xi)|$') plt.xlabel('x') plt.ylabel(r'$\xi$') plt.title("Micro-Support Estimate (1/|Symbol|)") plt.show() def visualize_phase(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0)-
Plot the phase (argument) of the pseudodifferential operator's symbol p(x, ξ) or p(x, y, ξ, η).
This visualization helps in understanding the oscillatory behavior and regularity properties of the operator in phase space. The phase is displayed modulo 2π using a cyclic colormap ('twilight') to emphasize its periodic nature.
Parameters
x_grid:ndarray- 1D array of spatial coordinates (x).
xi_grid:ndarray- 1D array of frequency coordinates (ξ).
y_grid:ndarray, optional- 2D spatial grid for y-coordinate (in 2D problems). Default is None.
eta_grid:ndarray, optional- 2D frequency grid for η (in 2D problems). Not used directly but kept for API consistency.
xi0:float, optional- Fixed value of ξ for slicing in 2D visualization. Default is 0.0.
eta0:float, optional- Fixed value of η for slicing in 2D visualization. Default is 0.0.
Notes: - In 1D: Displays arg(p(x, ξ)) over the (x, ξ) phase plane. - In 2D: Displays arg(p(x, y, ξ₀, η₀)) for fixed frequency values (ξ₀, η₀). - Uses plt.pcolormesh with 'twilight' colormap to represent angles from -π to π.
Raises: - NotImplementedError: If the spatial dimension is not 1D or 2D.
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def visualize_phase(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Plot the phase (argument) of the pseudodifferential operator's symbol p(x, ξ) or p(x, y, ξ, η). This visualization helps in understanding the oscillatory behavior and regularity properties of the operator in phase space. The phase is displayed modulo 2π using a cyclic colormap ('twilight') to emphasize its periodic nature. Parameters ---------- x_grid : ndarray 1D array of spatial coordinates (x). xi_grid : ndarray 1D array of frequency coordinates (ξ). y_grid : ndarray, optional 2D spatial grid for y-coordinate (in 2D problems). Default is None. eta_grid : ndarray, optional 2D frequency grid for η (in 2D problems). Not used directly but kept for API consistency. xi0 : float, optional Fixed value of ξ for slicing in 2D visualization. Default is 0.0. eta0 : float, optional Fixed value of η for slicing in 2D visualization. Default is 0.0. Notes: - In 1D: Displays arg(p(x, ξ)) over the (x, ξ) phase plane. - In 2D: Displays arg(p(x, y, ξ₀, η₀)) for fixed frequency values (ξ₀, η₀). - Uses plt.pcolormesh with 'twilight' colormap to represent angles from -π to π. Raises: - NotImplementedError: If the spatial dimension is not 1D or 2D. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.pcolormesh(X, XI, np.angle(symbol_vals), shading='auto', cmap='twilight') plt.colorbar(label='arg(Symbol) [rad]') plt.xlabel('x') plt.ylabel('ξ') plt.title('Phase Portrait (arg p(x, ξ))') plt.show() elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.pcolormesh(X, Y, np.angle(symbol_vals), shading='auto', cmap='twilight') plt.colorbar(label='arg(Symbol) [rad]') plt.xlabel('x') plt.ylabel('y') plt.title(f'Phase Portrait at ξ={xi0}, η={eta0}') plt.show() def visualize_symbol_amplitude(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0)-
Display the modulus |p(x, ξ)| or |p(x, y, ξ₀, η₀)| as a color map.
This method visualizes the amplitude of the pseudodifferential operator's symbol in either 1D or 2D spatial configuration. In 2D, the frequency variables are fixed to specified values (ξ₀, η₀) for visualization purposes.
Parameters
x_grid,y_grid:ndarray- Spatial grids over which to evaluate the symbol. y_grid is optional and used only in 2D.
xi_grid,eta_grid:ndarray- Frequency grids. In 2D, these define the domain over which the symbol is evaluated, but the visualization fixes ξ = ξ₀ and η = η₀.
xi0,eta0:float, optional- Fixed frequency values for slicing in 2D visualization. Defaults to zero.
Notes
- In 1D: Visualizes |p(x, ξ)| over the (x, ξ) grid.
- In 2D: Visualizes |p(x, y, ξ₀, η₀)| at fixed frequencies ξ₀ and η₀.
- The color intensity represents the magnitude of the symbol, highlighting regions where the symbol is large or small.
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def visualize_symbol_amplitude(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Display the modulus |p(x, ξ)| or |p(x, y, ξ₀, η₀)| as a color map. This method visualizes the amplitude of the pseudodifferential operator's symbol in either 1D or 2D spatial configuration. In 2D, the frequency variables are fixed to specified values (ξ₀, η₀) for visualization purposes. Parameters ---------- x_grid, y_grid : ndarray Spatial grids over which to evaluate the symbol. y_grid is optional and used only in 2D. xi_grid, eta_grid : ndarray Frequency grids. In 2D, these define the domain over which the symbol is evaluated, but the visualization fixes ξ = ξ₀ and η = η₀. xi0, eta0 : float, optional Fixed frequency values for slicing in 2D visualization. Defaults to zero. Notes ----- - In 1D: Visualizes |p(x, ξ)| over the (x, ξ) grid. - In 2D: Visualizes |p(x, y, ξ₀, η₀)| at fixed frequencies ξ₀ and η₀. - The color intensity represents the magnitude of the symbol, highlighting regions where the symbol is large or small. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid, indexing='ij') symbol_vals = self.p_func(X, XI) plt.pcolormesh(X, XI, np.abs(symbol_vals), shading='auto') plt.colorbar(label='|Symbol|') plt.xlabel('x') plt.ylabel('ξ') plt.title('Symbol Amplitude |p(x, ξ)|') plt.show() elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid, indexing='ij') XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.pcolormesh(X, Y, np.abs(symbol_vals), shading='auto') plt.colorbar(label='|Symbol|') plt.xlabel('x') plt.ylabel('y') plt.title(f'Symbol Amplitude at ξ={xi0}, η={eta0}') plt.show() def visualize_wavefront(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0)-
Visualize the wavefront set of the pseudo-differential operator's symbol by plotting |p(x, ξ)| or |p(x, y, ξ₀, η₀)|.
The wavefront set captures the location and direction of singularities in the symbol p, which is crucial in microlocal analysis for understanding propagation of singularities and wave behavior.
In 1D, this method displays |p(x, ξ)| over the phase space (x, ξ). In 2D, it fixes the frequency values (ξ₀, η₀) and visualizes |p(x, y, ξ₀, η₀)| over the spatial domain to highlight where the symbol interacts with the fixed frequency.
Parameters
x_grid:ndarray- 1D array of spatial coordinates (x) used for evaluation.
xi_grid:ndarray- 1D array of frequency coordinates (ξ) used for visualization in 1D or slicing in 2D.
y_grid:ndarray, optional- 1D array of second spatial coordinate (y), used only in 2D.
eta_grid:ndarray, optional- 1D array of second frequency coordinate (η), not directly used but kept for API consistency.
xi0:float, default=0.0- Fixed value of ξ for slicing in 2D visualization.
eta0:float, default=0.0- Fixed value of η for slicing in 2D visualization.
Notes
- In 1D: Displays a 2D color map of |p(x, ξ)| over the phase space with axes (x, ξ).
- In 2D: Evaluates and plots |p(x, y, ξ₀, η₀)| at fixed frequencies (ξ₀, η₀) over the spatial grid (x, y).
- Uses
imshowfor fast and scalable visualization with automatic aspect ratio adjustment. - A colormap ('viridis') is used to enhance contrast and readability of the magnitude variations.
Expand source code
def visualize_wavefront(self, x_grid, xi_grid, y_grid=None, eta_grid=None, xi0=0.0, eta0=0.0): """ Visualize the wavefront set of the pseudo-differential operator's symbol by plotting |p(x, ξ)| or |p(x, y, ξ₀, η₀)|. The wavefront set captures the location and direction of singularities in the symbol p, which is crucial in microlocal analysis for understanding propagation of singularities and wave behavior. In 1D, this method displays |p(x, ξ)| over the phase space (x, ξ). In 2D, it fixes the frequency values (ξ₀, η₀) and visualizes |p(x, y, ξ₀, η₀)| over the spatial domain to highlight where the symbol interacts with the fixed frequency. Parameters ---------- x_grid : ndarray 1D array of spatial coordinates (x) used for evaluation. xi_grid : ndarray 1D array of frequency coordinates (ξ) used for visualization in 1D or slicing in 2D. y_grid : ndarray, optional 1D array of second spatial coordinate (y), used only in 2D. eta_grid : ndarray, optional 1D array of second frequency coordinate (η), not directly used but kept for API consistency. xi0 : float, default=0.0 Fixed value of ξ for slicing in 2D visualization. eta0 : float, default=0.0 Fixed value of η for slicing in 2D visualization. Notes ----- - In 1D: Displays a 2D color map of |p(x, ξ)| over the phase space with axes (x, ξ). - In 2D: Evaluates and plots |p(x, y, ξ₀, η₀)| at fixed frequencies (ξ₀, η₀) over the spatial grid (x, y). - Uses `imshow` for fast and scalable visualization with automatic aspect ratio adjustment. - A colormap ('viridis') is used to enhance contrast and readability of the magnitude variations. """ if self.dim == 1: X, XI = np.meshgrid(x_grid, xi_grid) symbol_vals = self.p_func(X, XI) plt.imshow(np.abs(symbol_vals), extent=[xi_grid.min(), xi_grid.max(), x_grid.min(), x_grid.max()], aspect='auto', origin='lower', cmap='viridis') elif self.dim == 2: X, Y = np.meshgrid(x_grid, y_grid) XI = np.full_like(X, xi0) ETA = np.full_like(Y, eta0) symbol_vals = self.p_func(X, Y, XI, ETA) plt.imshow(np.abs(symbol_vals), extent=[x_grid.min(), x_grid.max(), y_grid.min(), y_grid.max()], aspect='auto', origin='lower', cmap='viridis') plt.colorbar(label='|Symbol|') plt.xlabel('ξ/Position') plt.ylabel('η/Position') plt.title('Wavefront Set') plt.show()
class psiOp (*args)-
Symbolic wrapper for PseudoDifferentialOperator. Usage: psiOp(symbol_expr, u)
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class psiOp(Function): """Symbolic wrapper for PseudoDifferentialOperator. Usage: psiOp(symbol_expr, u) """ nargs = 2 # (expr, u)Ancestors
- sympy.core.function.Function
- sympy.core.function.Application
- sympy.core.expr.Expr
- sympy.core.basic.Basic
- sympy.printing.defaults.Printable
- sympy.core.evalf.EvalfMixin
Class variables
var default_assumptionsvar nargs