An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

TL;DR

This paper introduces an asymptotic preserving numerical scheme for the nonlinear Schrödinger equation in the semiclassical limit, utilizing a novel phase/amplitude formulation with vanishing viscosity to ensure stability and accuracy.

Contribution

It presents a new phase/amplitude formulation with vanishing viscosity and a second order asymptotic preserving scheme for the NLS in the semiclassical regime, ensuring robustness before singularities.

Findings

The system is locally well-posed in Sobolev spaces.

The scheme is globally well-posed in 1D for fixed positive Planck constant.

The numerical method accurately recovers physical observables independent of mesh size and Planck constant.

Abstract

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.