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Journal Articles IMA Journal of Numerical Analysis Year : 2020

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

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 $s\geq3$.
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Dates and versions

hal-01656394 , version 1 (05-12-2017)
hal-01656394 , version 2 (12-12-2017)
hal-01656394 , version 3 (29-10-2018)

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Clémentine Courtès, Frédéric Lagoutière, Frédéric Rousset. Error estimates of finite difference schemes for the Korteweg-de Vries equation. IMA Journal of Numerical Analysis, 2020, 40 (1), pp.628-685. ⟨10.1093/imanum/dry082⟩. ⟨hal-01656394v3⟩
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