2.19. Kuramoto-Sivashinsky - Using custom class

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

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

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from pde import PDEBase, ScalarField, UnitGrid


class KuramotoSivashinskyPDE(PDEBase):
    """Implementation of the normalized Kuramoto–Sivashinsky equation"""

    def evolution_rate(self, state, t=0):
        """implement the python version of the evolution equation"""
        state_lap = state.laplace(bc="auto_periodic_neumann")
        state_lap2 = state_lap.laplace(bc="auto_periodic_neumann")
        state_grad = state.gradient(bc="auto_periodic_neumann")
        return -state_grad.to_scalar("squared_sum") / 2 - state_lap - state_lap2


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

eq = KuramotoSivashinskyPDE()  # define the pde
result = eq.solve(state, t_range=10, dt=0.01)
result.plot()