An asymptotically preserving scheme for nonlinear Schrodinger equation in the semiclassical limit

Rémi Carles (1) , Bijan Mohammadi (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

We study numerically the semiclassical limit for the nonlinear Schrodinger equation thanks to a modification of the Madelung transform due to E.Grenier. This approach is naturally asymptotically preserving. 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.

Domaines: Analyse numérique [math.NA]

Rémi Carles, Bijan Mohammadi. An asymptotically preserving scheme for nonlinear Schrodinger equation in the semiclassical limit. 2010. ⟨hal-00454427v1⟩