Error Estimate for a Fully Discrete Spectral Scheme for Korteweg-de Vries-Kawahara Equation

TL;DR

This paper analyzes the convergence and error bounds of a fully discrete spectral scheme combining Fourier-Galerkin in space and leap-frog in time for the Kawahara equation, relevant in fluid dynamics.

Contribution

It provides the first rigorous error estimate for a fully discrete spectral method applied to the Kawahara equation, demonstrating spectral accuracy in space and second-order in time.

Findings

Spectral accuracy in space achieved

Second-order temporal accuracy proven

Error bounds established for the fully discrete scheme

Abstract

We are concerned with the convergence of spectral method for the numerical solution of 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 fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier- Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.