Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation

TL;DR

This paper proves the convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, covering both decaying and periodic cases, with results depending on the initial data's regularity.

Contribution

It establishes convergence of a finite difference scheme for the Korteweg-de Vries equation in both classical and weak solution frameworks based on initial data regularity.

Findings

Convergence to classical solutions for high regularity initial data.

Strong convergence to weak solutions for lower regularity initial data.

Applicable to both decaying and periodic boundary conditions.

Abstract

We prove convergence of a fully discrete finite difference scheme for the Korteweg--de Vries equation. Both the decaying case on the full line and the periodic case are considered. If the initial data $u|_{t=0}=u_0$ is of high regularity, $u_0\in H^3(\R)$, the scheme is shown to converge to a classical solution, and if the regularity of the initial data is smaller, $u_0\in L^2(\R)$, then the scheme converges strongly in $L^2(0,T;L^2_{\mathrm{loc}}(\R))$ to a weak solution.