Discrete energy estimates for the abcd-systems
This work provides rigorous numerical analysis for a class of water wave models, but the contribution is incremental as it extends existing finite volume techniques to specific parameter regimes.
The paper proposes finite volume schemes for abcd-systems and establishes stability and error estimates, achieving O(Δt+(Δx)^2) accuracy when bd>0 and O(Δt+Δx) when bd=0. Numerical experiments validate the theoretical results and investigate head-on collisions of traveling waves.
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 $O(Δt+(Δx)^2)$ accurate, while if $bd=0$, we obtain an $O(Δt+Δx)$ -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.