2.2.4 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:

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

  0%|          | 0/10.0 [00:00<?, ?it/s]
Initializing:   0%|          | 0/10.0 [00:00<?, ?it/s]
  0%|          | 0/10.0 [00:10<?, ?it/s]
  0%|          | 0.01/10.0 [00:10<2:56:13, 1058.41s/it]
  2%|▏         | 0.21/10.0 [00:10<08:13, 50.40s/it]
  2%|▏         | 0.21/10.0 [00:10<08:13, 50.42s/it]
100%|██████████| 10.0/10.0 [00:10<00:00,  1.06s/it]
100%|██████████| 10.0/10.0 [00:10<00:00,  1.06s/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()