Error estimates of finite difference schemes for the Korteweg-de Vries equation

TL;DR

This paper analyzes the error estimates of finite difference schemes for the Korteweg-de Vries equation, proving convergence rates under various CFL conditions and demonstrating near-optimal accuracy through numerical simulations.

Contribution

It establishes convergence and error bounds for a finite difference scheme applied to the Korteweg-de Vries equation, including non-smooth initial data cases.

Findings

First order convergence for smooth solutions in Sobolev space $H^s$ with $s \\geq 6$

Extension of convergence results to less smooth initial data with $s \\geq 3/4$

Numerical simulations suggest the convergence orders are near optimal for $s \\geq 3$

Abstract

This article deals with the numerical analysis of the Cauchy problem for the Korteweg-de Vries equation with a finite difference scheme. We consider the Rusanov scheme for the hyperbolic flux term and a 4-points $\theta$-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant-Friedrichs-Lewy condition when $\theta\geq \frac{1}{2}$ and under an "Airy" Courant-Friedrichs-Lewy condition when $\theta<\frac{1}{2}$. More precisely, we get the first order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the non-smooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$ , to the price of a loss in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq3$.