We study numerically the semiclassical limit for the nonlinear
Schrödinger equation thanks to a modification of the Madelung
transform due to Grenier. This approach 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.