Numerical analysis with error estimates for the Korteweg-de Vries Equation

Clémentine Courtès (1) , Frédéric Lagoutière (2) , Frédéric Rousset (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 ICJ - Institut Camille Jordan

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. The orders of convergence seem optimal with numerical simulations in some cases, at least when $s\geq3$.

Mots clés: finite difference scheme; numerical convergence; Korteweg-de Vries equation; error estimates

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Clémentine Courtès, Frédéric Lagoutière, Frédéric Rousset. Numerical analysis with error estimates for the Korteweg-de Vries Equation. 2017. ⟨hal-01656394v1⟩