Communications in Mathematical Sciences
Volume 17 (2019)
Discrete energy estimates for the $abcd$-systems
Pages: 243 – 298
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
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