  We study numerically the semiclassical limit for the nonlinear Schroedinger
equation thanks to a modification of the Madelung transform due to E.Grenier.
This approach is naturally asymptotic preserving, and allows for the presence
of vacuum. Even if the mesh size and the time step do not depend on the Planck
constant, we recover the position and current densities in the semiclassical
limit, with a numerical rate of convergence in accordance with the theoretical
results, before shocks appear in the limiting Euler equation. By using simple
projections, the mass and the momentum of the solution are well preserved by
the numerical scheme, while the variation of the energy is not negligible
numerically. Experiments suggest that beyond the critical time for the Euler
equation, Grenier's approach yields smooth but highly oscillatory terms.
