  We consider the semiclassical limit for the nonlinear Schrodinger equation.
We introduce a phase/amplitude representation given by a system similar to the
hydrodynamical formulation, whose novelty consists in including some
asymptotically vanishing viscosity. We prove that the system is always locally
well-posed in a class of Sobolev spaces, and globally well-posed for a fixed
positive Planck constant in the one-dimensional case. We propose a second order
numerical scheme which is asymptotic preserving. Before singularities appear in
the limiting Euler equation, we recover the quadratic physical observables as
well as the wave function with mesh size and time step independent of the
Planck constant. This approach is also well suited to the linear Schrodinger
equation.
