A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence

Liviu I. Ignat; Enrique Zuazua

doi:10.1016/j.crma.2005.07.018

Numerical Analysis/Partial Differential Equations

A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence

Presented by: 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 341 (2005) no. 6, pp. 381-386.

Abstract
Résumé

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.

On introduit une méthode bi-maille semi-discrète en différences finies pour l'approximation numérique de l'équation de Schrödinger. On démontre la convergence $L^{2}$ 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 l'algorithme bi-maille agit comme un filtre des hautes fréquences. On montre aussi la convergence du schéma dans une classe d'équations non-linéaires dont l'étude 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 l'aide d'un schéma avec viscosité numérique.

Received: 2005-03-25
Accepted: 2005-07-19
Published online: 2005-09-02

DOI: 10.1016/j.crma.2005.07.018

Author's affiliations:

Liviu I. Ignat ¹; Enrique Zuazua ¹

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

@article{CRMATH_2005__341_6_381_0,
     author = {Liviu I. Ignat and Enrique Zuazua},
     title = {A two-grid approximation scheme for nonlinear {Schr\"odinger} equations: dispersive properties and convergence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {381--386},
     publisher = {Elsevier},
     volume = {341},
     number = {6},
     year = {2005},
     doi = {10.1016/j.crma.2005.07.018},
     language = {en},
}

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Liviu I. Ignat; Enrique Zuazua. A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. Comptes Rendus. Mathématique, Volume 341 (2005) no. 6, pp. 381-386. doi : 10.1016/j.crma.2005.07.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.07.018/

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