Convergence of a higher-order scheme for Korteweg-de Vries equation

TL;DR

This paper develops and analyzes a higher-order Galerkin scheme for the Korteweg-de Vries equation, proving its convergence to weak solutions for L^2 initial data and demonstrating its effectiveness through examples.

Contribution

It introduces a new higher-order Galerkin scheme for the KdV equation and proves its convergence in L^2, advancing numerical methods for this nonlinear PDE.

Findings

Scheme achieves higher spatial accuracy and first-order temporal accuracy.

Convergence to weak solutions is established for L^2 initial data.

Numerical examples illustrate the scheme's effectiveness.

Abstract

We study the convergence of higher order schemes for the Cauchy problem associated to the KdV equation. More precisely, we design a Galerkin type implicit scheme which has higher order accuracy in space and first order accuracy in time. The convergence is established for initial data in L^2, and we show that the scheme converges strongly in L^2(0,T; L^2_loc(\R)) to a weak solution. Finally, the convergence is illustrated by several examples.