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

Christophe Besse; Rémi Carles; Florian Méhats

hal-00752011, version 1

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

Christophe Besse (, ) ^1, 2, Rémi Carles (, ) ³, Florian Méhats (, ) ^4, 5

Résumé : 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.

1 : Laboratoire Paul Painlevé (LPP)
CNRS : UMR8524 – Université Lille I - Sciences et technologies
2 : SIMPAF (INRIA Lille - Nord Europe)
INRIA – Université Lille I - Sciences et technologies – CNRS : UMR
3 : Institut de Mathématiques et de Modélisation de Montpellier (I3M)
CNRS : UMR5149 – Université Montpellier II - Sciences et techniques
4 : Institut de Recherche Mathématique de Rennes (IRMAR)
CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
5 : IPSO (INRIA - IRMAR)
CNRS : UMR6074 – INRIA – Université de Rennes 1

hal-00752011, version 1
http://hal.archives-ouvertes.fr/hal-00752011
oai:hal.archives-ouvertes.fr:hal-00752011
Contributeur : Rémi Carles <>
Soumis le : Mercredi 14 Novembre 2012, 16:16:27
Dernière modification le : Jeudi 15 Novembre 2012, 09:42:52

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arXiv : 1211.3391

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