Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Discrete energy estimates for the $abcd$-systems

Pages: 243 – 298



Cosmin Burtea (Université Paris-Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Unité Mixte de Recherche (UMR) Paris, France)

Clémentine Courtès (Institut de Mathématiques de Toulouse, UMR, Université de Toulouse, Centre National de la Recherche Scientifique, Institut National des Sciences Appliquées, Toulouse, France)


In this article, we propose finite volume schemes for the $abcd$-systems and we establish stability and error estimates. The order of accuracy depends on the so-called BBM-type dispersion coefficients $b$ and $d$. If $bd \gt 0$, the numerical schemes are $O(\Delta t + (\Delta x)^2)$ accurate, while if $bd=0$, we obtain an $O(\Delta t + \Delta x)$ order of convergence. The analysis covers a broad range of the parameters $a,b,c,d$. In the second part of the paper, numerical experiments validating the theoretical results as well as head-on collision of traveling waves are investigated.


system $abcd$, numerical convergence, error estimates

2010 Mathematics Subject Classification

35Q35, 65M12

This work was performed within the framework of the LABEX MILYON (ANR-10- LABX0070) of Université de Lyon, and within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Received 8 January 2018

Accepted 16 November 2018

Published 30 May 2019