2.6. Kuramoto-Sivashinsky - Using PDE class

This example implements a scalar PDE using the PDE. We here consider the Kuramoto–Sivashinsky equation, which for instance describes the dynamics of flame fronts:

\[\partial_t u = -\frac12 |\nabla u|^2 - \nabla^2 u - \nabla^4 u\]

Out:

  0%|          | 0/10.0 [00:00<?, ?it/s]
Initializing:   0%|          | 0/10.0 [00:00<?, ?it/s]
  0%|          | 0/10.0 [00:12<?, ?it/s]
  0%|          | 0.01/10.0 [00:25<6:58:47, 2515.30s/it]
  0%|          | 0.02/10.0 [00:27<3:52:02, 1395.06s/it]
  0%|          | 0.03/10.0 [00:27<2:34:32, 930.05s/it]
  1%|1         | 0.1/10.0 [00:27<46:02, 279.02s/it]
 44%|####4     | 4.4/10.0 [00:27<00:35,  6.34s/it]
 44%|####4     | 4.4/10.0 [00:27<00:35,  6.35s/it]
100%|##########| 10.0/10.0 [00:27<00:00,  2.79s/it]
100%|##########| 10.0/10.0 [00:27<00:00,  2.79s/it]

from pde import PDE, ScalarField, UnitGrid

grid = UnitGrid([32, 32])  # generate grid
state = ScalarField.random_uniform(grid)  # generate initial condition

eq = PDE({"u": "-gradient_squared(u) / 2 - laplace(u + laplace(u))"})  # define the pde
result = eq.solve(state, t_range=10, dt=0.01)
result.plot()