Notes on a high order fully discrete scheme for the Korteweg-de vries equation with a time-stepping procedure of Runge-Kutta-composition type

TL;DR

This paper develops and analyzes a high-order, fully discrete numerical scheme for the periodic Korteweg-de Vries equation, combining spectral Fourier-Galerkin spatial discretization with an implicit Runge-Kutta time-stepping method, and provides error estimates.

Contribution

It introduces a novel high-order, fully discrete scheme for the KdV equation using spectral and Runge-Kutta methods, with proven error bounds.

Findings

Established $L^{2}$ error estimates for the scheme.

Demonstrated the scheme's high-order accuracy.

Provided rigorous convergence analysis.

Abstract

We consider the periodic initial-value problem for the Korteweg-de Vries equation that we discretize in space by a spectral Fourier-Galerkin method and in time by an implicit, high order, Runge-Kutta scheme of composition type based on the implicit midpoint rule. We prove $L^{2}$ error estimates for the resulting semidiscrete and the fully discrete approximations.