  In this paper, we prove that the time supremum of the Wasserstein distance
between the time-marginals of a uniformly elliptic multidimensional diffusion
with coefficients bounded together with their derivatives up to the order $2$
in the spatial variables and H{\"o}lder continuous with exponent $\gamma$ with
respect to the time variable and its Euler scheme with $N$ uniform time-steps
is smaller than $C \left(1+\mathbf{1}\_{\gamma=1}
\sqrt{\ln(N)}\right)N^{-\gamma}$. To do so, we use the theory of optimal
transport. More precisely, we investigate how to apply the theory by Ambrosio,
Gigli and Savar{\'e} to compute the time derivative of the Wasserstein distance
between the time-marginals. We deduce a stability inequality for the
Wasserstein distance which finally leads to the desired estimation.
