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

We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter $\epsilon$. 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 $\epsilon$. The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case $P({\mathrm{\partial }}_x)={\mathrm{\partial }}_x$ 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 $\epsilon$. In regularizing regimes $\epsilon =\mathcal{O}(1)$ we even gain a derivative with our time discretisation at the cost of loosing in terms of $\frac{1}{\epsilon }$. Numerical experiments underline our theoretical findings.