Discrete energy estimates for the abcd-systems

TL;DR

This paper develops finite volume schemes for the abcd-systems, providing stability and error estimates that depend on dispersion coefficients, and validates the results through numerical experiments including wave collisions.

Contribution

It introduces finite volume schemes for abcd-systems with comprehensive stability and error analysis across various parameters.

Findings

Schemes achieve $O( riangle t + ( riangle x)^2)$ accuracy when $bd>0$.

Schemes achieve $O( riangle t + riangle x)$ accuracy when $bd=0$.

Numerical experiments confirm theoretical error estimates and simulate wave collisions.

Abstract

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 $bd>0$, 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.