In [1]:
%reload_ext autoreload
%autoreload 2
In [2]:
from PDESolver import *
1D Examples in periodic conditions¶
1D Complex Ginzburg-Landau equation¶
In [3]:
# 1. Define the symbols and the equation
t, x = symbols('t x', real=True)
u = Function('u', complex=True)
# Equation parameters
alpha = 1.0
beta = 0.5 # Imaginary part of diffusion
gamma = 1.0
delta = 0.2 # Imaginary part of the linear term
epsilon = 0.1 # Imaginary part of the nonlinearity
# Complex Ginzburg-Landau equation
linear_diffusion = (alpha + I*beta) * diff(u(t, x), x, x)
linear_growth = (gamma + I*delta) * u(t, x)
nonlinear_term = (1 + I*epsilon) * Abs(u(t, x))**2 * u(t, x)
# Complete equation
equation = Eq(diff(u(t, x), t), linear_diffusion + linear_growth - nonlinear_term)
# 2. Initial condition: complex plane wave
def initial_condition(x_vals):
A = 1.0 # Amplitude
k = 0.5 # Wave number
return A * np.exp(1j * k * x_vals) # Exact solution in the absence of nonlinearity
# 3. Solver setup
solver = PDESolver(equation, time_scheme='ETD-RK4', dealiasing_ratio=2/3)
# Simulation parameters
Lx = 50.0
Nx = 1024
Lt = 10.0
Nt = 1000
# Setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='periodic',
initial_condition=initial_condition
)
# 4. Solve
solver.solve()
# 5. Visualization
ani = solver.animate(component='abs')
HTML(ani.to_jshtml())
********************************* * Partial differential equation * *********************************
2
∂ 2 ∂
──(u(t, x)) = - (1 + 0.1⋅ⅈ)⋅u(t, x)⋅│u(t, x)│ + (1.0 + 0.2⋅ⅈ)⋅u(t, x) + (1.0 + 0.5⋅ⅈ)⋅───(u(t, x))
∂t 2
∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: (1 + 0.1*I)*u(t, x)*Abs(u(t, x))**2 - (1.0 + 0.2*I)*u(t, x) + Derivative(u(t, x), t) - (1.0 + 0.5*I)*Derivative(u(t, x), (x, 2))
Expanded equation: u(t, x)*Abs(u(t, x))**2 + 0.1*I*u(t, x)*Abs(u(t, x))**2 - 1.0*u(t, x) - 0.2*I*u(t, x) + Derivative(u(t, x), t) - 1.0*Derivative(u(t, x), (x, 2)) - 0.5*I*Derivative(u(t, x), (x, 2))
Analyzing term: u(t, x)*Abs(u(t, x))**2
--> Classified as nonlinear
Analyzing term: 0.1*I*u(t, x)*Abs(u(t, x))**2
--> Classified as nonlinear
Analyzing term: -1.0*u(t, x)
--> Classified as linear
Analyzing term: -0.2*I*u(t, x)
--> Classified as linear
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Analyzing term: -1.0*Derivative(u(t, x), (x, 2))
Derivative found: Derivative(u(t, x), (x, 2))
--> Classified as linear
Analyzing term: -0.5*I*Derivative(u(t, x), (x, 2))
Derivative found: Derivative(u(t, x), (x, 2))
--> Classified as linear
Final linear terms: {u(t, x): 0, Derivative(u(t, x), t): 1, Derivative(u(t, x), (x, 2)): -1.0 - 0.5*I}
Final nonlinear terms: [u(t, x)*Abs(u(t, x))**2, 0.1*I*u(t, x)*Abs(u(t, x))**2]
Symbol terms: []
Pseudo terms: []
Source terms: []
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
2
kx ⋅(1.0 + 0.5⋅ⅈ) - ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
2
kx ⋅(1.0 + 0.5⋅ⅈ) - ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = []
--- Solutions found ---
⎡ 2 ⎤
⎣kx ⋅(0.5 - ⅈ)⎦
--- Final linear operator ---
2
-ⅈ⋅kx ⋅(0.5 - ⅈ)
*****************
* CFL condition *
*****************
CFL condition violated: dt = 0.01, max allowed dt = 1.1795338201608715e-05
********************
* Symbol condition *
********************
✅ Spectral stability satisfied: Re(a(k)) ≤ 0
✅ Proper high-frequency dissipation
✅ Reasonable spectral growth
✔ Symbol analysis completed.
*******************
* Symbol plotting *
*******************
******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:4234: RuntimeWarning: invalid value encountered in divide return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1) / z, 1.0) /home/fifi/pdesolver_prod/PDESolver.py:4237: RuntimeWarning: invalid value encountered in divide return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1 - z) / z**2, 0.5)
********************* * Solution plotting * *********************
/home/fifi/anaconda3/lib/python3.12/site-packages/matplotlib/cbook.py:1709: ComplexWarning: Casting complex values to real discards the imaginary part return math.isfinite(val) /home/fifi/anaconda3/lib/python3.12/site-packages/matplotlib/transforms.py:2875: ComplexWarning: Casting complex values to real discards the imaginary part vmin, vmax = map(float, [vmin, vmax])
Out[3]:
1D Sine-Gordon equation¶
In [4]:
# 1. Define the symbols and the equation
t, x = symbols('t x', real=True)
u = Function('u', real=True)
# Sine-Gordon equation
equation = Eq(diff(u(t, x), t, t), diff(u(t, x), x, x) + sin(u(t, x)))
# 2. Initial condition: stationary kink
def initial_condition(x_vals):
x0 = 0 # Position of the kink
return 4 * np.arctan(np.exp(x_vals - x0)) # Kink soliton
# 3. Initial velocity (for a second-order equation in time)
def initial_velocity(x_vals):
return np.zeros_like(x_vals) # Initial velocity is zero
# 4. Solver configuration
solver = PDESolver(equation, time_scheme='ETD-RK4')
# Simulation parameters
Lx = 20.0
Nx = 512
Lt = 40.0
Nt = 1500
# Setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='periodic',
initial_condition=initial_condition,
initial_velocity=initial_velocity # Necessary for second-order equations
)
# 5. Solve
solver.solve()
# 6. Visualization
ani = solver.animate(component='abs') # Visualize the amplitude of u
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
2 2
∂ ∂
───(u(t, x)) = sin(u(t, x)) + ───(u(t, x))
2 2
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: -sin(u(t, x)) + Derivative(u(t, x), (t, 2)) - Derivative(u(t, x), (x, 2))
Expanded equation: -sin(u(t, x)) + Derivative(u(t, x), (t, 2)) - Derivative(u(t, x), (x, 2))
Analyzing term: -sin(u(t, x))
--> Classified as nonlinear
Analyzing term: Derivative(u(t, x), (t, 2))
Derivative found: Derivative(u(t, x), (t, 2))
--> Classified as linear
Analyzing term: -Derivative(u(t, x), (x, 2))
Derivative found: Derivative(u(t, x), (x, 2))
--> Classified as linear
Final linear terms: {Derivative(u(t, x), (t, 2)): 1, Derivative(u(t, x), (x, 2)): -1}
Final nonlinear terms: [-sin(u(t, x))]
Symbol terms: []
Pseudo terms: []
Source terms: []
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
2 2
kx - ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
2 2
kx - ω
Temporal order from dispersion relation: 2
self.pseudo_terms = []
--- Solutions found ---
[-kx, kx]
--- Final linear operator ---
2
-kx
*****************
* CFL condition *
*****************
CFL condition violated: dt = 0.02666666666666667, max allowed dt = 0.019492953724695495
********************
* Symbol condition *
********************
✅ Spectral stability satisfied: Re(a(k)) ≤ 0
✅ Proper high-frequency dissipation
✅ Reasonable spectral growth
✔ Symbol analysis completed.
*******************
* Symbol plotting *
*******************
***************************** * Wave propagation analysis * *****************************
******************* * Solving the PDE * *******************
********************* * Solution plotting * *********************
Out[4]:
1D Kuramoto–Sivashinsky equation¶
In [5]:
# 1. Symbols
t, x, kx = symbols('t x kx')
u_func = Function('u')
u = u_func(t, x)
# 2. Kuramoto–Sivashinsky equation
# eq = Eq(diff(u, t), -u * diff(u, x) - diff(u, x, 2) - diff(u, x, 4))
eq = Eq(diff(u, t), -u * diff(u, x) - diff(u, x, 2) - 0.1 * diff(u, x, 4))
# 3. Creating the solver
solver = PDESolver(eq, time_scheme='ETD-RK4', dealiasing_ratio=0.5)
# 4. Domain parameters
Lx = 2 * np.pi
Nx = 256
Lt = 50.0
Nt = 1000
# 5. Initial condition: smooth random perturbation
rng = np.random.default_rng(42)
# def initial_condition(x_vals):
# return 0.1 * np.sin(x_vals) + 0.01 * rng.standard_normal(x_vals.shape)
def initial_condition(x_vals):
return 0.1 * np.cos(3 * x_vals) + 0.01 * rng.normal(size=x_vals.shape)
# 6. Setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='periodic',
initial_condition=initial_condition
)
# 7. Solving
solver.solve()
# 8. Visualization
ani = solver.animate(component='real')
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
2 4
∂ ∂ ∂ ∂
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) - ───(u(t, x)) - 0.1⋅───(u(t, x))
∂t ∂x 2 4
∂x ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) + Derivative(u(t, x), t) + Derivative(u(t, x), (x, 2)) + 0.1*Derivative(u(t, x), (x, 4))
Expanded equation: u(t, x)*Derivative(u(t, x), x) + Derivative(u(t, x), t) + Derivative(u(t, x), (x, 2)) + 0.1*Derivative(u(t, x), (x, 4))
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Analyzing term: Derivative(u(t, x), (x, 2))
Derivative found: Derivative(u(t, x), (x, 2))
--> Classified as linear
Analyzing term: 0.1*Derivative(u(t, x), (x, 4))
Derivative found: Derivative(u(t, x), (x, 4))
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1, Derivative(u(t, x), (x, 2)): 1, Derivative(u(t, x), (x, 4)): 0.100000000000000}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: []
Source terms: []
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
4 2
0.1⋅kx - kx - ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
4 2
0.1⋅kx - kx - ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = []
--- Solutions found ---
⎡ 2 ⎛ 2⎞⎤
⎣0.1⋅ⅈ⋅kx ⋅⎝10.0 - kx ⎠⎦
--- Final linear operator ---
2 ⎛ 2⎞
0.1⋅kx ⋅⎝10.0 - kx ⎠
*****************
* CFL condition *
*****************
********************
* Symbol condition *
********************
❌ Stability violated: max Re(a(k)) = 2.499731395159199
Unstable symbol: Re(a(k)) > 0
⚠️ Insufficient high-frequency dissipation
✅ Reasonable spectral growth
✔ Symbol analysis completed.
*******************
* Symbol plotting *
*******************
******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:4234: RuntimeWarning: invalid value encountered in divide return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1) / z, 1.0) /home/fifi/pdesolver_prod/PDESolver.py:4237: RuntimeWarning: invalid value encountered in divide return np.where(np.abs(z) > 1e-12, (np.exp(z) - 1 - z) / z**2, 0.5)
********************* * Solution plotting * *********************
Out[5]:
1D Examples in Dirichlet conditions¶
1D non-linear Schrödinger equation¶
In [6]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: -∂²/∂x² → ψOp(xi²)
linear_symbol = xi**2
# Non-linear term: |u|² * u
nonlinear_term = Abs(u(t, x))**2 * u(t, x)
# Non-linear Schrödinger equation
equation = Eq(I * diff(u(t, x), t), psiOp(linear_symbol, u(t, x)) + nonlinear_term)
# Create the solver
solver = PDESolver(equation)
# Parameters
Lx = 20.0
Nx = 512
Lt = 10.0
Nt = 1000
# Initial condition: fundamental soliton
def initial_condition(x):
return 2 / np.cosh(x) # Order 1 soliton
# Configuration
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
# Solve
solver.solve()
# Visualization
ani = solver.animate(component='abs')
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
∂ 2 ⎛ 2 ⎞
ⅈ⋅──(u(t, x)) = u(t, x)⋅│u(t, x)│ + psiOp⎝ξ , u(t, x)⎠
∂t
********************
* Equation parsing *
********************
Equation rewritten in standard form: -u(t, x)*Abs(u(t, x))**2 - psiOp(xi**2, u(t, x)) + I*Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: -u(t, x)*Abs(u(t, x))**2 - psiOp(xi**2, u(t, x)) + I*Derivative(u(t, x), t)
Analyzing term: -u(t, x)*Abs(u(t, x))**2
--> Classified as nonlinear
Analyzing term: -psiOp(xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: I*Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): I}
Final nonlinear terms: [-u(t, x)*Abs(u(t, x))**2]
Symbol terms: []
Pseudo terms: [(-1, xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, xi**2)]
✅ Time derivative coefficient detected: I
symbol =
2
-ⅈ⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
********************* * Solution plotting * *********************
Out[6]:
1D Heat equation with source¶
In [7]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: ∂²/∂x² → ψOp(-xi²)
linear_symbol = -xi**2
# Heat source: Gaussian pulse centered at x=0
source_expr = exp(-(x**2) - t**2)
# Heat equation with source
equation = Eq(diff(u(t, x), t), psiOp(linear_symbol, u(t, x)) + source_expr)
# Solver
solver = PDESolver(equation)
# Spatial and temporal domain
Lx = 10.0
Nx = 256
Lt = 5.0
Nt = 500
# Initialization: zero temperature initially
def initial_condition(x):
return np.zeros_like(x)
# Dirichlet boundary conditions: fixed temperature at the ends
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
# Solving
solver.solve()
# Visualization
ani = solver.animate()
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
2 2
∂ - t - x ⎛ 2 ⎞
──(u(t, x)) = ℯ + psiOp⎝-ξ , u(t, x)⎠
∂t
********************
* Equation parsing *
********************
Equation rewritten in standard form: -exp(-t**2 - x**2) - psiOp(-xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: -exp(-t**2 - x**2) - psiOp(-xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: -exp(-t**2 - x**2)
--> Classified as source term
Analyzing term: -psiOp(-xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: []
Symbol terms: []
Pseudo terms: [(-1, -xi**2)]
Source terms: [-exp(-t**2 - x**2)]
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
********************* * Solution plotting * *********************
/home/fifi/pdesolver_prod/PDESolver.py:4962: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding. ax.set_ylim(np.min(self.frames[0]), np.max(self.frames[0]))
/home/fifi/pdesolver_prod/PDESolver.py:4974: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding. ax.set_ylim(np.min(ydata_real), np.max(ydata_real))
Out[7]:
1D KdV equation¶
In [8]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: ∂³/∂x³ → ψOp(i ξ³)
linear_symbol = I * xi**3
# Nonlinear term: u ∂u/∂x
nonlinear_term = u(t, x) * diff(u(t, x), x)
# KdV equation
equation = Eq(diff(u(t, x), t), psiOp(linear_symbol, u(t, x)) - nonlinear_term)
# Solver
solver = PDESolver(equation)
# Parameters
Lx = 40.0
Nx = 512
Lt = 30.0
Nt = 1500
# Initial condition: KdV soliton
def initial_condition(x):
return 3 * (1 / np.cosh(x / 2))**2 # Height 3, width 2
# Setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
# Solve
solver.solve()
# Visualization
ani = solver.animate(component='real')
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
∂ ∂ ⎛ 3 ⎞
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp⎝ⅈ⋅ξ , u(t, x)⎠
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(I*xi**3, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(I*xi**3, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(I*xi**3, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, I*xi**3)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, I*xi**3)]
✅ Time derivative coefficient detected: 1
symbol =
3
ⅈ⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
********************* * Solution plotting * *********************
Out[8]:
1D Airy equation (not sure that it works - low-amplitude linear waves in shallow water...)¶
In [9]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: ∂³/∂x³ → ψOp(i ξ³)
linear_symbol = I * xi**3
# Airy equation
equation = Eq(diff(u(t, x), t), psiOp(linear_symbol, u(t, x)))
# Solver
solver = PDESolver(equation)
# Parameters
Lx = 40.0
Nx = 512
Lt = 10.0
Nt = 500
# Initial condition: localized pulse (deformed Gaussian)
def initial_condition(x):
return np.exp(-x**2) * (1 - 2*x**2)
# Configuration
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
# Solving
solver.solve()
# Visualization
ani = solver.animate()
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
∂ ⎛ 3 ⎞
──(u(t, x)) = psiOp⎝ⅈ⋅ξ , u(t, x)⎠
∂t
********************
* Equation parsing *
********************
Equation rewritten in standard form: -psiOp(I*xi**3, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: -psiOp(I*xi**3, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: -psiOp(I*xi**3, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: []
Symbol terms: []
Pseudo terms: [(-1, I*xi**3)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, I*xi**3)]
✅ Time derivative coefficient detected: 1
symbol =
3
ⅈ⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * ******************* ********************* * Solution plotting * *********************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
Out[9]:
1D Fisher-KPP equation¶
In [10]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: D ∂²/∂x² → ψOp(-D ξ²)
D = 0.5
linear_symbol = -D * xi**2
# Non-linear term: r u(1 - u), where r = growth rate
r = 1.0
nonlinear_term = r * u(t, x) * (1 - u(t, x))
# Fisher-KPP equation
equation = Eq(diff(u(t, x), t), psiOp(linear_symbol, u(t, x)) + nonlinear_term)
# Creation of the solver
solver = PDESolver(equation)
# Parameters
Lx = 20.0
Nx = 256
Lt = 10.0
Nt = 500
# Initial condition: a wave localized at the center
def initial_condition(x):
return np.where(np.abs(x) < 1, 1.0, 0.0)
# Configuration
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
# Solving
solver.solve()
# Visualization
ani = solver.animate()
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
∂ ⎛ 2 ⎞
──(u(t, x)) = 1.0⋅(1 - u(t, x))⋅u(t, x) + psiOp⎝-0.5⋅ξ , u(t, x)⎠
∂t
********************
* Equation parsing *
********************
Equation rewritten in standard form: -1.0*(1 - u(t, x))*u(t, x) - psiOp(-0.5*xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: -1.0*(1 - u(t, x))*u(t, x) - psiOp(-0.5*xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: -1.0*(1 - u(t, x))*u(t, x)
--> Classified as linear
Analyzing term: -psiOp(-0.5*xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {u(t, x): 1, Derivative(u(t, x), t): 1}
Final nonlinear terms: []
Symbol terms: []
Pseudo terms: [(-1, -0.5*xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω + 1
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω + 1
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -0.5*xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-0.5⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis.
*******************
* Solving the PDE *
*******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
********************* * Solution plotting * *********************
Out[10]:
1D Klein-Gordon equation with Higgs potential¶
In [11]:
# Symbolic variables
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Linear operator: ∂²/∂x² → ψOp(-ξ²)
mass = 1.0
nonlinear_coeff = 0.5
linear_symbol = -xi**2 - mass**2
# Non-linear term: λ u³
nonlinear_term = nonlinear_coeff * u(t, x)**3
# Klein-Gordon equation with Higgs potential
equation = Eq(diff(u(t, x), t, 2), psiOp(linear_symbol, u(t, x)) + nonlinear_term)
# Solver
solver = PDESolver(equation)
# Spatial and temporal domain
Lx = 20.0
Nx = 512
Lt = 20.0
Nt = 1000
# Initial condition: localized wave (Gaussian)
def initial_condition(x):
return np.exp(-(x)**2 / 0.5)
# Initial velocity is zero
def initial_velocity(x):
return np.zeros_like(x)
# Configuration
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition,
initial_velocity=initial_velocity
)
# Solving
solver.solve()
# Visualization
ani = solver.animate()
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
2
∂ 3 ⎛ 2 ⎞
───(u(t, x)) = 0.5⋅u (t, x) + psiOp⎝- ξ - 1.0, u(t, x)⎠
2
∂t
********************
* Equation parsing *
********************
Equation rewritten in standard form: -0.5*u(t, x)**3 - psiOp(-xi**2 - 1.0, u(t, x)) + Derivative(u(t, x), (t, 2))
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: -0.5*u(t, x)**3 - psiOp(-xi**2 - 1.0, u(t, x)) + Derivative(u(t, x), (t, 2))
Analyzing term: -0.5*u(t, x)**3
--> Classified as nonlinear
Analyzing term: -psiOp(-xi**2 - 1.0, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), (t, 2))
Derivative found: Derivative(u(t, x), (t, 2))
--> Classified as linear
Final linear terms: {Derivative(u(t, x), (t, 2)): 1}
Final nonlinear terms: [-0.5*u(t, x)**3]
Symbol terms: []
Pseudo terms: [(-1, -xi**2 - 1.0)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
2
-ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
2
-ω
Temporal order from dispersion relation: 2
self.pseudo_terms = [(-1, -xi**2 - 1.0)]
✅ Time derivative coefficient detected: 1
symbol =
2
- ξ - 1.0
⚠️ For psiOp, use interactive_symbol_analysis.
*******************
* Solving the PDE *
*******************
********************* * Solution plotting * *********************
Out[11]:
1D Benjamin–Ono equation¶
In [12]:
# Symbols
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
# Benjamin–Ono Equation: ∂ₜ u = ψOp(–i |ξ|, u) – u ∂ₓ u
symbol = -I * Abs(xi)
nonlinear = u(t, x) * diff(u(t, x), x)
equation = Eq(diff(u(t, x), t), psiOp(symbol, u(t, x)) - nonlinear)
# Initial condition: soliton-like bump
def initial_condition(x):
return 2 / (1 + x**2)
# Solver configuration
solver = PDESolver(equation, time_scheme='ETD-RK4', dealiasing_ratio=2/3)
# Parameters
Lx = 40.0
Nx = 1024
Lt = 30.0
Nt = 2000
# Setup
solver.setup(
Lx=Lx,
Nx=Nx,
Lt=Lt,
Nt=Nt,
initial_condition=initial_condition,
boundary_condition='dirichlet'
)
# Solving
solver.solve()
# Visualization
ani = solver.animate(component='real')
HTML(ani.to_jshtml())
*********************************
* Partial differential equation *
*********************************
∂ ∂
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp(-ⅈ⋅│ξ│, u(t, x))
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(-I*Abs(xi), u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(-I*Abs(xi), u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(-I*Abs(xi), u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, -I*Abs(xi))]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -I*Abs(xi))]
✅ Time derivative coefficient detected: 1
symbol =
-ⅈ⋅│ξ│
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
********************* * Solution plotting * *********************
Out[12]:
1D Burgers' equation: Comparative Animation of Viscosity Effects¶
In [13]:
t, x, xi = symbols('t x xi', real=True)
u = Function('u')
fig, ax = plt.subplots()
colors = ["r", "g", "b", "k"]
lines = []
## Viscosity: 4 values
nus = [1, 0.1, 0.01, 0.001]
# --- Spatio-temporal domain ---
Lx = 3 * np.pi
Nx = 512
Lt = 100
Nt = 2000
# --- Some initial conditions ---
def ic_sine(x):
"""Simple sine wave"""
return np.sin(x)
def ic_gaussian(x, x0=0.0, sigma=np.sqrt(0.1)):
return np.exp(-0.5 * ((x - x0) / sigma) ** 2)
def ic_sech2(x, x0=0.0, delta=0.5, A=2.0):
"""A / cosh^2((x-x0)/delta) (sech^2)"""
return A / np.cosh((x - x0) / delta) ** 2
def ic_tanh_shock(x, x0=0.0, k=5.0, nu=0.1):
"""Regularized shock profile (tanh)"""
return -2 * nu * k * np.tanh(0.5 * k * (x - x0))
def ic_two_pulses(x):
"""Two opposing Gaussians (collision)"""
return ic_gaussian(x, x0=1, sigma=0.5) - ic_gaussian(x, x0=-1, sigma=0.5)
def ic_multimode(x, modes=10, amp_decay=1.0):
"""Sum of sines with random phases -> nonlinear cascade"""
rng = np.random.RandomState(12345) # reproducible
u0 = np.zeros_like(x)
for k in range(1, modes+1):
phase = rng.uniform(0, 2*np.pi)
u0 += (amp_decay / k) * np.sin(k * x + phase)
return u0
# Choice
initial_condition = ic_two_pulses
for nu, c in zip(nus, colors):
## PDE definition using pseudodifferential approach (due to Dirichlet condition).
linear_symbol = -nu * xi**2
nonlinear_term = u(t, x) * diff(u(t, x), x)
eq = Eq(diff(u(t, x), t), psiOp(linear_symbol, u(t, x)) - nonlinear_term)
solver = PDESolver(eq)
solver.setup(
Lx=Lx, Nx=Nx, Lt=Lt, Nt=Nt,
boundary_condition='dirichlet',
initial_condition=initial_condition
)
sol = np.array(solver.solve())
xgrid = np.linspace(-Lx/2, Lx/2, Nx)
line, = ax.plot([], [], c, label=f"nu={nu}")
lines.append((line, sol, xgrid))
ax.set_title(r"Burgers' equation: $u_t + u u_x = \nu u_{xx}$")
ax.legend()
ax.set_xlim(-Lx/2, Lx/2)
ax.set_ylim(-1, 1)
dt = Lt / Nt # time increment
def init():
for line, _, _ in lines:
line.set_data([], [])
return [l for l, _, _ in lines]
def update(frame):
t_val = frame * dt
for (line, sol, xgrid) in lines:
line.set_data(xgrid, sol[frame, :])
ax.set_xlabel(f"t= {t_val:.2f}")
return [l for l, _, _ in lines]
ani = FuncAnimation(fig, update, frames=len(lines[0][1]), init_func=init, blit=True)
display(HTML(ani.to_jshtml()))
*********************************
* Partial differential equation *
*********************************
∂ ∂ ⎛ 2 ⎞
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp⎝-ξ , u(t, x)⎠
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(-xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(-xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(-xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, -xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-ξ
⚠️ For psiOp, use interactive_symbol_analysis.
*******************
* Solving the PDE *
*******************
/home/fifi/pdesolver_prod/PDESolver.py:3539: RuntimeWarning: invalid value encountered in divide phi1_L = (exp_L - 1.0) / (self.dt * L_vals)
*********************************
* Partial differential equation *
*********************************
∂ ∂ ⎛ 2 ⎞
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp⎝-0.1⋅ξ , u(t, x)⎠
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.1*xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.1*xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(-0.1*xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, -0.1*xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -0.1*xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-0.1⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis.
*******************
* Solving the PDE *
*******************
*********************************
* Partial differential equation *
*********************************
∂ ∂ ⎛ 2 ⎞
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp⎝-0.01⋅ξ , u(t, x)⎠
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.01*xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.01*xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(-0.01*xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, -0.01*xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -0.01*xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-0.01⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
*********************************
* Partial differential equation *
*********************************
∂ ∂ ⎛ 2 ⎞
──(u(t, x)) = - u(t, x)⋅──(u(t, x)) + psiOp⎝-0.001⋅ξ , u(t, x)⎠
∂t ∂x
********************
* Equation parsing *
********************
Equation rewritten in standard form: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.001*xi**2, u(t, x)) + Derivative(u(t, x), t)
⚠️ psiOp detected: skipping expansion for safety
Expanded equation: u(t, x)*Derivative(u(t, x), x) - psiOp(-0.001*xi**2, u(t, x)) + Derivative(u(t, x), t)
Analyzing term: u(t, x)*Derivative(u(t, x), x)
--> Classified as nonlinear
Analyzing term: -psiOp(-0.001*xi**2, u(t, x))
--> Classified as pseudo linear term (psiOp)
Analyzing term: Derivative(u(t, x), t)
Derivative found: Derivative(u(t, x), t)
--> Classified as linear
Final linear terms: {Derivative(u(t, x), t): 1}
Final nonlinear terms: [u(t, x)*Derivative(u(t, x), x)]
Symbol terms: []
Pseudo terms: [(-1, -0.001*xi**2)]
Source terms: []
⚠️ Pseudo‑differential operator detected: all other linear terms have been rejected.
*******************************
* Linear operator computation *
*******************************
Characteristic equation before symbol treatment:
-ⅈ⋅ω
--- Symbolic symbol analysis ---
symb_omega: 0
symb_k: 0
Raw characteristic equation:
-ⅈ⋅ω
Temporal order from dispersion relation: 1
self.pseudo_terms = [(-1, -0.001*xi**2)]
✅ Time derivative coefficient detected: 1
symbol =
2
-0.001⋅ξ
⚠️ For psiOp, use interactive_symbol_analysis. ******************* * Solving the PDE * *******************
/home/fifi/anaconda3/lib/python3.12/site-packages/matplotlib/cbook.py:1345: ComplexWarning: Casting complex values to real discards the imaginary part return np.asarray(x, float)
In [ ]: