KdV.md

Solving the KdV Equation using DiffEqOperators

In this tutorial we will try to solve the famous KdV equation which describes the motion of waves on shallow water surfaces. The equation is commonly written as

    $$\frac{\partial u}{\partial t} + \frac{{\partial}^3 u}{\partial x^3} - 6u\frac{\partial u}{\partial x} = 0$$.

Let's consider the cosine wave as the initial waveform and evolve it using the equation with a periodic boundary condition. This example is taken from here.

using DiffEqOperators, DifferentialEquations, Plots
x = collect(0 : 1/99 : 2);
u0 = cos.(π*x);
du3 = zeros(u0); # a container array
function KdV(t, u, du)
    C(t,u,du3)
    A(t, u, du)
    copy!(du, -u.*du .- 0.022^2.*du3)
end

Now defining our DiffEqOperators:

A = DerivativeOperator{Float64}(1,2,1/99,199,:periodic,:periodic);
C = DerivativeOperator{Float64}(3,2,1/99,199,:periodic,:periodic);

Now call the ODE solver as follows:

prob1 = ODEProblem(KdV, u0, (0.,5.));
sol1 = solve(prob1, dense=false, tstops=0:0.01:10);

and plot the solutions to see the waveform evolve with time. Note: The waveform being solved for here is non-directional unlike the many waves you might see like traveling solitons. In that case you might need to use the Upwind operators.