One dimension Korteweg–de Vries study case.

The Korteweg–de Vries equation is a non-linear PDE modeling wave on shallow water surfaces. It reads

\[\frac{\partial U}{\partial t} + U\,\frac{\partial U}{\partial x} = a\,\frac{\partial^2 U}{\partial x^2} + b\,\frac{\partial^3 U}{\partial x^3}\]

The initial conditions is taken as a smoothed triangle. The discontinuity occuring usually in Burger equation results here in a train of capillary wave after the wave front.

This example can be compeared with this Dedalus Project example where the same model is solved with pseudo-spectral method.

import numpy as np
import pylab as pl
from skfdiff import Model, Simulation

model = Model("-U * dxU + a * dxxU + b * dxxxU", "U(x)", ["a", "b"])

x = np.linspace(-2, 6, 1000)

n = 20
U = np.log(1 + np.cosh(n) ** 2 / np.cosh(n * x) ** 2) / (2 * n)

initial_fields = model.fields_template(x=x, U=U, a=2e-4, b=1e-4)

simulation = Simulation(model, initial_fields, dt=0.05, tmax=10)
container = simulation.attach_container()

simulation.run()
(
    container.data.U[: -2 : container.data.t.size // 6].plot(
        col="t", col_wrap=3, color="black"
    )
)
pl.show()