Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameter

TL;DR

This paper introduces new splitting schemes for the Benjamin-Bona-Mahony equation that are uniformly accurate across dispersive parameters, preserving the KdV limit and providing improved error bounds, especially for small dispersive regimes.

Contribution

The authors develop and analyze a class of splitting methods that are uniformly accurate in the dispersive parameter and preserve the KdV limit, with rigorous convergence proofs and improved error estimates.

Findings

Schemes are uniformly accurate in dispersive parameter 

First-order convergence in $H^r$ without regularity loss for classical BBM

Numerical experiments confirm theoretical error bounds

Abstract

We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter $\varepsilon$. The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter $\varepsilon$. The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case $P(\partial_x) = \partial_x$ our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in $H^{r}$ for solutions in $H^{r}$ without any loss of derivative. This estimate holds uniformly in $\varepsilon$. In regularizing regimes $\varepsilon=\mathcal{O}(1) $ we even gain a derivative with our time discretisation at the cost of loosing in terms of $\frac{1}{\varepsilon}$. Numerical experiments underline our theoretical findings.