  We give bounds for the total variation distance between the solutions to two
stochastic differential equations starting at the same point and with close
coefficients, which applies in particular to the distance between an exact
solution and its Euler-Maruyama scheme in small time. We show that for small
$t$, the total variation distance is of order $t^{r/(2r+1)}$ if the noise
coefficient $\sigma$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in
\mathbb{N}$ and if the drift is $C^1$ with bounded derivatives, using
multi-step Richardson-Romberg extrapolation. We do not require the drift to be
bounded. Then we prove with a counterexample that we cannot achieve a bound
better than $t^{1/2}$ in general.
