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Comptes Rendus Mathématique
Volume 345, n° 9
pages 531-536 (novembre 2007)
Doi : 10.1016/j.crma.2007.10.014
Received : 5 September 2007 ;  accepted : 24 September 2007
An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit
Un schéma AP pour lʼéquation de Schrödinger dans la limite semi-classique
 

Pierre Degond a , Samy Gallego a , Florian Méhats b
a IMT, Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France 
b IRMAR, Université de Rennes, campus de Beaulieu, 35042 Rennes cedex, France 

Abstract

This Note is devoted to the discretization of the fluid formulation of the Schrödinger equation (the Madelung system). We explore both the discretization of the system in Eulerian coordinates and 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 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 tends to 0. Numerical simulations are given; they confirm that the considered schemes allow us to numerically bridge the quantum and semi-classical scales. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.
Résumé

Cette Note est consacrée à la discrétisation de la formulation fluide de lʼéquation de Schrödinger (le système de Madelung) en formulations eulerienne et lagrangienne. Nous proposons des schémas pour ces deux formulations qui sont implicites dans le terme de flux de masse. Cette caractéristique nous permet de montrer que ces schémas sont asymptotiquement préservatifs, cʼest à dire quʼils fournissent une discrétisation des équations de Hamilton-Jacobi semi-classiques lorsque la constante de Planck adimensionnée tend vers 0. De plus, une analyse linéarisée permet de montrer que ces schémas sont asymptotiquement stables, cʼest à dire que leur contrainte de stabilité reste bornée lorsque tend vers 0. Des simulations numériques sont proposées ; elles confirment que les schémas considérés permettent de fournir une passerelle numérique entre les échelles quantiques et semi-classiques. Pour citer cet article : P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.


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