Dispersive properties of a viscous numerical scheme for the Schrödinger equation

Liviu I. Ignat; Enrique Zuazua

doi:10.1016/j.crma.2005.02.024

Numerical Analysis/Partial Differential Equations

Dispersive properties of a viscous numerical scheme for the Schrödinger equation
[Propriétés dispersives d'un schéma numérique visqueux pour l'équation de Schrödinger]

Présenté par : Pierre-Louis Lions

Liviu I. Ignat ¹ ; Enrique Zuazua ¹

¹ Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 340 (2005) no. 7, pp. 529-534.

Résumé
Abstract

Dans cette Note on étudié les propriétés dispersives des schémas d'approximation numérique de l'équation de Schrödinger. On considère des approximations semi-discretes en différences finies. Nous démontrons d'abord que le schéma conservatif habituel ne reproduit pas les propriétés dispersives, uniformement par rapport au pas du maillage. Ceci est du aux hautes fréquences numériques artificielles. On introduit donc un schéma d'approximation visqueux dissipant ces hautes fréquences et l'on montre qu'il possède des propriétés de dispersivité uniformes par rapport au pas du maillage. Nous appliquons ce schéma à l'approximation numérique des équations de Schrödinger non-linéaires. On démontre la convergence dans la classe de non-linéarités dont l'analyse, au niveau de l'équation de Schrödinger continue, a besoin des inegalités de Strichartz.

In this Note we study the dispersive properties of the numerical approximation schemes for the free Schrödinger equation. We consider finite-difference space semi-discretizations. We first show that the standard conservative scheme does not reproduce at the discrete level the properties of the continuous Schrödinger equation. This is due to spurious high frequency numerical solutions. In order to damp out these high-frequencies and to reflect the properties of the continuous problem we add a suitable extra numerical viscosity term at a convenient scale. We prove that the dispersive properties of this viscous scheme are uniform when the mesh-size tends to zero. Finally we prove the convergence of this viscous numerical scheme for a class of nonlinear Schrödinger equations with nonlinearities that may not be handeled by standard energy methods and that require the so-called Strichartz inequalities.

Reçu le : 2005-01-03
Accepté le : 2005-02-21
Publié le : 2005-03-23

DOI : 10.1016/j.crma.2005.02.024

Affiliations des auteurs :

Liviu I. Ignat ¹ ; Enrique Zuazua ¹

¹ Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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     author = {Liviu I. Ignat and Enrique Zuazua},
     title = {Dispersive properties of a viscous numerical scheme for the {Schr\"odinger} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {529--534},
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     number = {7},
     year = {2005},
     doi = {10.1016/j.crma.2005.02.024},
     language = {en},
}

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Liviu I. Ignat; Enrique Zuazua. Dispersive properties of a viscous numerical scheme for the Schrödinger equation. Comptes Rendus. Mathématique, Volume 340 (2005) no. 7, pp. 529-534. doi : 10.1016/j.crma.2005.02.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.02.024/

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