Burgers Equation¶

Burgers equation:

$$ \frac{\partial \mathbf{u}}{\partial t} =-\mathbf{u} \cdot \nabla \mathbf{u} + \nu \nabla^2 \mathbf{u} $$

In [1]:

from torchfsm.operator import Operator, Convection, Laplacian

def Burgers(nu:float) -> Operator:
    return nu*Laplacian()-Convection()
burgers=Burgers(0.01)

from torchfsm.operator import Operator, Convection, Laplacian def Burgers(nu:float) -> Operator: return nu*Laplacian()-Convection() burgers=Burgers(0.01)

In [2]:

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;

Solution in 1D¶

In [3]:

mesh=MeshGrid([(0,L,N)],device=device)
x=mesh.bc_mesh_grid()
u_0=torch.sin(2*torch.pi*x/L)+0.5
traj=burgers.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=200,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N)],device=device) x=mesh.bc_mesh_grid() u_0=torch.sin(2*torch.pi*x/L)+0.5 traj=burgers.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)

We also randomly generate a batch of initial conditions and solve the Burgers equation in a batched manner:

In [17]:

from torchfsm.field import wave_1d
x=mesh.bc_mesh_grid(batch_size=3)
u_0=wave_1d(x,
            batched=True,
            min_k=2,max_k=3,
            min_amplitude=0.1,
            max_amplitude=0.5,
            mean_shift_coef=2)
traj=burgers.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.005,
    step=200,
    trajectory_recorder=AutoRecorder(),
)

from torchfsm.field import wave_1d x=mesh.bc_mesh_grid(batch_size=3) u_0=wave_1d(x, batched=True, min_k=2,max_k=3, min_amplitude=0.1, max_amplitude=0.5, mean_shift_coef=2) traj=burgers.integrate( u_0=u_0, mesh=mesh, dt=0.005, step=200, trajectory_recorder=AutoRecorder(), )

In [18]:

plot_traj(traj,animation=False)

plot_traj(traj,animation=False)

Solution in 2D¶

In [5]:

mesh=MeshGrid([(0,L,N),(0,L,N)],device=device)
x,y=mesh.bc_mesh_grid()
u_0=torch.cat([torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L),
               torch.cos(2*torch.pi*x/L)*torch.sin(4*torch.pi*y/L)],
              dim=1)
traj=burgers.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N),(0,L,N)],device=device) x,y=mesh.bc_mesh_grid() u_0=torch.cat([torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L), torch.cos(2*torch.pi*x/L)*torch.sin(4*torch.pi*y/L)], dim=1) traj=burgers.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), )

In [6]:

plot_traj(traj)

plot_traj(traj)

Out[6]:

Solution in 3D¶

In [7]:

mesh=MeshGrid([(0,L,N),(0,L,N),(0,L,N)],device=device)
x,y,z=mesh.bc_mesh_grid()
u_0=torch.cat([torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L),
               torch.cos(2*torch.pi*x/L)*torch.sin(4*torch.pi*y/L)*torch.cos(6*torch.pi*z/L),
               torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L)],
              dim=1)
traj=burgers.integrate(
    u_0=u_0,
    mesh=mesh,
    dt=0.01,
    step=50,
    trajectory_recorder=AutoRecorder(),
)

mesh=MeshGrid([(0,L,N),(0,L,N),(0,L,N)],device=device) x,y,z=mesh.bc_mesh_grid() u_0=torch.cat([torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L), torch.cos(2*torch.pi*x/L)*torch.sin(4*torch.pi*y/L)*torch.cos(6*torch.pi*z/L), torch.sin(2*torch.pi*x/L)*torch.cos(4*torch.pi*y/L)*torch.sin(6*torch.pi*z/L)], dim=1) traj=burgers.integrate( u_0=u_0, mesh=mesh, dt=0.01, step=50, trajectory_recorder=AutoRecorder(), )

In [8]:

plot_traj(traj)

plot_traj(traj)

Out[8]: