Swift-Hohenberg Equation¶

The Swift–Hohenberg equation is a partial differential equation noted for its pattern-forming behaviour. It is defined as:

$$ \frac{\partial \phi}{\partial t} = r \phi - (1 + \Delta)^2 \phi + \phi^2 - \phi^3 $$

Solution in 1D¶

In [1]:

from torchfsm.operator import Operator, Laplacian, ImplicitSource, HyperDiffusion

def SwiftHohenberg(r:float) -> Operator:
    return -HyperDiffusion()-2*Laplacian()+ImplicitSource(lambda phi:r*phi-phi+phi**2-phi**3)

swift_hohenberg = SwiftHohenberg(r=2)

from torchfsm.operator import Operator, Laplacian, ImplicitSource, HyperDiffusion def SwiftHohenberg(r:float) -> Operator: return -HyperDiffusion()-2*Laplacian()+ImplicitSource(lambda phi:r*phi-phi+phi**2-phi**3) swift_hohenberg = SwiftHohenberg(r=2)

In [2]:

import torch
from torchfsm.mesh import MeshGrid
from torchfsm.traj_recorder import AutoRecorder
from torchfsm.plot import plot_traj
from torchfsm.field import truncated_fourier_series
device='cuda' if torch.cuda.is_available() else 'cpu'
L=20.0*torch.pi; N=100; 

import torch from torchfsm.mesh import MeshGrid from torchfsm.traj_recorder import AutoRecorder from torchfsm.plot import plot_traj from torchfsm.field import truncated_fourier_series device='cuda' if torch.cuda.is_available() else 'cpu' L=20.0*torch.pi; N=100;

In [3]:

mesh=MeshGrid([(0,L,N)],device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=10,
    normalize_mode="-1_1", 
)
traj=swift_hohenberg.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.1,
    step=200,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N)],device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=10, normalize_mode="-1_1", ) traj=swift_hohenberg.integrate( u_0=u_0, mesh=mesh, dt=0.1, step=200, trajectory_recorder=AutoRecorder(), )

In [4]:

plot_traj(traj,animation=False)

plot_traj(traj,animation=False)

Solution in 2D¶

In [5]:

mesh=MeshGrid([(0,L,N)]*2,device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=10,
    normalize_mode="-1_1", 
)
traj=swift_hohenberg.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.1,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N)]*2,device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=10, normalize_mode="-1_1", ) traj=swift_hohenberg.integrate( u_0=u_0, mesh=mesh, dt=0.1, step=50, trajectory_recorder=AutoRecorder(), )

In [6]:

plot_traj(traj,animation=True)

plot_traj(traj,animation=True)

Out[6]:

Solution in 3D¶

In [7]:

mesh=MeshGrid([(0,L,N)]*3,device=device)
x=mesh.bc_mesh_grid()
u_0=truncated_fourier_series(
    mesh=mesh,
    freq_threshold=10,
    normalize_mode="-1_1", 
)
traj=swift_hohenberg.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.1,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N)]*3,device=device) x=mesh.bc_mesh_grid() u_0=truncated_fourier_series( mesh=mesh, freq_threshold=10, normalize_mode="-1_1", ) traj=swift_hohenberg.integrate( u_0=u_0, mesh=mesh, dt=0.1, step=50, trajectory_recorder=AutoRecorder(), )

In [8]:

plot_traj(traj,animation=True,)

plot_traj(traj,animation=True,)

Out[8]: