An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit
Résumé
This note is devoted to the discretization of the fluid formulation of the Schrödinger equation (the Madelung system). We explore the discretization of the system both in Eulerian coordinates and in Lagrangian coordinates. We propose schemes for these two formulations which are implicit in the mass flux term. This feature allows us to show that these schemes are asymptotic preserving; i.e., they provide discretizations of the semi-classical Hamilton-Jacobi equation when the scaled Planck constant $\epsilon$ tends to 0. An analysis performed on the linearized systems also shows that they are asymptotically stable; i.e., their stability condition remains bounded as $\epsilon$ tends to 0. Numerical simulations are given; they confirm that the schemes considered allow us to numerically bridge the quantum and semi-classical scales