An asymptotic preserving approach for nonlinear Schrodinger equation in the semiclassical limit

TL;DR

This paper introduces an asymptotic preserving numerical method for the nonlinear Schrödinger equation in the semiclassical limit, effectively handling vacuum states and capturing the limiting behavior with controlled convergence before shock formation.

Contribution

It presents a modified Madelung transform-based approach that remains asymptotic preserving and accurately approximates position and current densities without mesh dependence on the Planck constant.

Findings

Accurate recovery of densities in the semiclassical limit

Mass and momentum are well preserved numerically

Energy variation is non-negligible in simulations

Abstract

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.