Finite Difference Schemes for the Korteweg-de Vries-Kawahara Equation

Abstract

We are concerned with the convergence of fully discrete finite difference schemes for the Korteweg-de Vries-Kawahara equation, which is a transport equation perturbed by dispersive terms of third and fifth order. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. Both the decaying case on the full line and the periodic case are considered. If the initial data $u|_{t=0} = u_0$ are of high regularity, $u_0\in H^5(\mathbb{R})$, the schemes are shown to converge to a classical solution. Finally, the convergence is illustrated by an example.