  For a stopped diffusion process in a multidimensional time-dependent domain
$\D$, we propose and analyse a new procedure consisting in simulating the
process with an Euler scheme with step size $\Delta$ and stopping it at
discrete times $(i\Delta)_{i\in\N^*}$ in a modified domain, whose boundary has
been appropriately shifted. The shift is locally in the direction of the inward
normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its
amplitude is equal to $0.5826 (...) |n^*\sigma|(t,x)\sqrt \Delta$ where
$\sigma$ stands for the diffusion coefficient of the process. The procedure is
thus extremely easy to use. In addition, we prove that the rate of convergence
w.r.t. $\Delta$ for the associated weak error is higher than without shifting,
generalizin g previous results by \cite{broa:glas:kou:97} obtained for the one
dimensional Brownian motion. For this, we establish in full generality the
asymptotics of the triplet exit time/exit position/overshoot for the discretely
stopped Euler scheme. Here, the overshoot means the distance to the boundary of
the process when it exits the domain. Numerical experiments support these
results.
