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Comptes Rendus Mathématique
Volume 341, n° 6
pages 381-386 (septembre 2005)
Doi : 10.1016/j.crma.2005.07.018
Received : 25 Mars 2005 ;  accepted : 19 July 2005
A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence
Un schéma de discrétisation bi-maille pour les équations de Schrödinger non-linéaires : propriétés dispersives et convergence
 

Liviu I. Ignat , Enrique Zuazua
Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain 

Abstract

We introduce a two-grid finite difference approximation scheme for the free Schrödinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schrödinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative to the method introduced by the authors [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529-534] using numerical viscosity. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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Résumé

On introduit une méthode bi-maille semi-discrète en différences finies pour lʼapproximation numérique de lʼéquation de Schrödinger. On démontre la convergence   du schéma et des propriétés dispersives uniformes par rapport au pas du maillage. Une analyse soigneuse en Fourier du symbole du schéma (consistant essentiellement à projeter des données lentes sur un maillage fin) montre que lʼalgorithme bi-maille agit comme un filtre des hautes fréquences. On montre aussi la convergence du schéma dans une classe dʼéquations non-linéaires dont lʼétude dans le cas continu nécessite des inégalités de Strichartz. Cette méthode donne une approche alternative à celle introduite par les auteurs [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529-534] à lʼaide dʼun schéma avec viscosité numérique. Pour citer cet article : L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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