  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.
