2.22 1D problem - Using custom class

This example implements a PDE that is only defined in one dimension. Here, we chose the Korteweg-de Vries equation, given by

\[\partial_t \phi = 6 \phi \partial_x \phi - \partial_x^3 \phi\]

which we implement using a custom PDE class below.

pde 1d class
from math import pi

from pde import CartesianGrid, MemoryStorage, PDEBase, ScalarField, plot_kymograph

class KortewegDeVriesPDE(PDEBase):
    """Korteweg-de Vries equation"""

    def evolution_rate(self, state, t=0):
        """implement the python version of the evolution equation"""
        assert state.grid.dim == 1  # ensure the state is one-dimensional
        grad_x = state.gradient("auto_periodic_neumann")[0]
        return 6 * state * grad_x - grad_x.laplace("auto_periodic_neumann")

# initialize the equation and the space
grid = CartesianGrid([[0, 2 * pi]], [32], periodic=True)
state = ScalarField.from_expression(grid, "sin(x)")

# solve the equation and store the trajectory
storage = MemoryStorage()
eq = KortewegDeVriesPDE()
eq.solve(state, t_range=3, solver="scipy", tracker=storage.tracker(0.1))

# plot the trajectory as a space-time plot

Total running time of the script: (0 minutes 2.425 seconds)