A comparison of second-order numerical methods applied to the Korteweg-de Vries equation

In recent years, the Korteweg-de Vries equation has been applied to many different phenomena such as collision-free hydromagnetic waves and longitudinal waves and it has been solved using many different numerical methods.

In this paper, we use a Fourier expansion method of approximately solving the Korteweg-de Vries equation, coupled with a choice of two second-order numerical methods for the time stepping procedure. That is, we generate a system of ordinary differential equations whose initial conditions are derived from the Fourier expansion of the initial values of the K dV equation and solve the problem using the trapezoidal method, which is implicit, and the leapfrog method, which is explicit. We study the errors involved with these methods, where we select model problems for which exact solutions are known. Also, we use the conservation laws associated with the K dV equation as another measure of the accuracy of the schemes.