Convergence of numerical schemes for the Korteweg-de Vries-Kawahara equation

TL;DR

This paper proves the convergence of semi-discrete and fully-discrete finite difference schemes for the Kawahara equation, a dispersive PDE modeling long waves in fluid dynamics, supported by numerical examples.

Contribution

It establishes the convergence of finite difference schemes for the Kawahara equation, a novel result for this dispersive PDE in fluid dynamics.

Findings

Convergence of semi-discrete schemes demonstrated.

Convergence of fully-discrete schemes demonstrated.

Numerical examples illustrate the theoretical results.

Abstract

We are concerned with the convergence of a numerical scheme for the initial-boundary value problem associated to the Korteweg-de Vries- Kawahara equation (in short Kawahara equation), which is a transport equation perturbed by dispersive terms of 3rd and 5th order. This equation appears in several uid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in uid dynamics. We prove here the convergence of both semi-discrete as well as fully-discrete finite difference schemes for the Kawahara equation. Finally, the convergence is illustratred by several examples.