Hyper Diffusion equation¶

The Hyper Diffusion equation is defined as:

$$ \frac{\partial \mathbf{u}}{\partial t} =\nu \nabla^4 \mathbf{u} $$

where $\nu$ is viscosity.

Solution in 1D¶

In [1]:

import torch
from torchfsm.mesh import MeshGrid
from torchfsm.plot import plot_traj
from torchfsm.traj_recorder import AutoRecorder
device='cuda' if torch.cuda.is_available() else 'cpu'
L=1.0; N=128; 

import torch from torchfsm.mesh import MeshGrid from torchfsm.plot import plot_traj from torchfsm.traj_recorder import AutoRecorder device='cuda' if torch.cuda.is_available() else 'cpu' L=1.0; N=128;

In [2]:

from torchfsm.operator import HyperDiffusion

from torchfsm.operator import HyperDiffusion

In [3]:

hyper_diffusion=-0.0001*HyperDiffusion()
mesh=MeshGrid([(0,L,N)],device=device)
x=mesh.bc_mesh_grid()
u_0=torch.where((x >0.3) & (x < 0.5), 1.0, 0.)
traj=hyper_diffusion.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=200,
    trajectory_recorder=AutoRecorder(),
)

hyper_diffusion=-0.0001*HyperDiffusion() mesh=MeshGrid([(0,L,N)],device=device) x=mesh.bc_mesh_grid() u_0=torch.where((x >0.3) & (x < 0.5), 1.0, 0.) traj=hyper_diffusion.integrate( u_0=u_0, mesh=mesh, dt=0.01, 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,y=mesh.bc_mesh_grid()
u_0=torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)
traj=hyper_diffusion.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N)]*2,device=device) x,y=mesh.bc_mesh_grid() u_0=torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L) traj=hyper_diffusion.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), )

In [6]:

plot_traj(traj,fps=10)

plot_traj(traj,fps=10)

Out[6]:

Solution in 3D¶

In [7]:

from torchfsm.plot import plot_traj, plot_field

from torchfsm.plot import plot_traj, plot_field

In [8]:

mesh=MeshGrid([(0,L,N)]*3,device=device)
x,y,z=mesh.bc_mesh_grid()
u_0=torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L)
traj=hyper_diffusion.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)
plot_traj(traj,fps=10)

mesh=MeshGrid([(0,L,N)]*3,device=device) x,y,z=mesh.bc_mesh_grid() u_0=torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L) traj=hyper_diffusion.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), ) plot_traj(traj,fps=10)

Out[8]: