Discrete energy estimates for the abcd-systems

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(∆t + (∆x)^2) accurate, while if bd = 0, we obtain an O(∆t + ∆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.