Burgers' equation is a partial differential equation consisting of an advection term and a diffusion term acting on a vector field $v$ (velocity). It reads
$$\frac{\partial v}{\partial t} = \nu \frac{\partial^2 v}{\partial x^2} - v \frac{\partial v}{\partial x}.$$In this example, we simulate Burgers' equation in 1D and 2D.
%pip install --quiet phiflow
from phi.jax.flow import *
# from phi.flow import * # If JAX is not installed. You can use phi.torch or phi.tf as well.
from tqdm.notebook import trange
We define a time step as consisting of diffusion and advection, according to the PDE above.
@jit_compile
def step(v, dt=.5):
v = diffuse.implicit(v, 0.1, dt=dt)
v = advect.semi_lagrangian(v, v, dt=dt)
return v
After setting the initial and boundary conditions, we repeatedly call step using the iterate shorthand.
v0 = CenteredGrid(Noise(smoothness=1.5), PERIODIC, x=64, bounds=Box(x=64))
v_trj = iterate(step, batch(time=100), v0)
plot(v_trj, animate='time')
By adding a y dimension to our v0 grid, we run the simulation in 2D.
v0 = CenteredGrid(Noise(vector='x,y'), PERIODIC, x=64, y=64, bounds=Box(x=40, y=20))
v_trj = iterate(step, batch(time=100), v0)
plot(v_trj.as_points(), animate='time')