3.1. Solve Differential Equations
Time Independent Problem¶
As discussed in the theory introduction, the Fourier spectral method can be used to solve time-independent equations that consist only of linear terms.
A classic example is Poisson’s equation:
$$ \nabla^2 \phi(\mathbf{x}) = s(\mathbf{x}), $$which can be solved in the Fourier domain as:
$$ \phi(\mathbf{x}) = \mathcal{F}^{-1}\left[\frac{\mathcal{F}[s(\mathbf{x})]}{(2\pi i \mathbf{f})^2}\right]. $$In TorchFSM, this is achieved using the solve function of a LinearOperator. The solve function takes a right-hand-side tensor and returns the corresponding solution tensor.
The following example demonstrates how to solve Poisson’s equation with a given source tensor.
from torchfsm.operator import Laplacian
from torchfsm.mesh import MeshGrid
from torchfsm.plot import plot_field
import torch
mesh=MeshGrid([(1,2*torch.pi,100)]*2)
x,y=mesh.bc_mesh_grid()
y=torch.cos(x)+torch.cos(y)
plot_field(y)
laplacian=Laplacian()
x=laplacian.solve(y,mesh=mesh,n_channel=1)
plot_field(laplacian.solve(y,mesh=mesh,n_channel=1))
Time-Dependent Problem¶
For a time-dependent problem of the form:
$$ \frac{\partial \mathbf{u}}{\partial t} = F(u), $$where $F(u)$ is a function that can be represented by an Operator, we can directly solve it using the integrate function of the Operator class.
The following example demonstrates how to solve a simple 1D diffusion problem:
from torchfsm.field import wave_1d
from torchfsm.mesh import MeshGrid
from torchfsm.operator import Laplacian
from torchfsm.traj_recorder import AutoRecorder
from torchfsm.plot import plot_traj
mesh=MeshGrid([(0,1,100)])
x=mesh.bc_mesh_grid()
diffusion=0.01*Laplacian()
u0=wave_1d(x)
us=diffusion.integrate(u0,dt=0.01,step=20,mesh=mesh,trajectory_recorder=AutoRecorder())
plot_traj(us,animation=True)
We will show more examples on the simulation of different PDEs in next section.