  We present a framework that allows for the non-asymptotic study of the
$2$-Wasserstein distance between the invariant distribution of an ergodic
stochastic differential equation and the distribution of its numerical
approximation in the strongly log-concave case. This allows us to study in a
unified way a number of different integrators proposed in the literature for
the overdamped and underdamped Langevin dynamics. In addition, we analyse a
novel splitting method for the underdamped Langevin dynamics which only
requires one gradient evaluation per time step. Under an additional smoothness
assumption on a $d$--dimensional strongly log-concave distribution with
condition number $\kappa$, the algorithm is shown to produce with an
$\mathcal{O}\big(\kappa^{5/4} d^{1/4}\epsilon^{-1/2} \big)$ complexity samples
from a distribution that, in Wasserstein distance, is at most $\epsilon>0$ away
from the target distribution.
