Comptes Rendus
no. 6
Numerical Analysis/Partial Differential Equations
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]

Présenté par : Pierre-Louis Lions

Liviu I. Ignat1 ; Enrique Zuazua1
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.

On introduit une méthode bi-maille semi-discrète en différences finies pour l'approximation numérique de l'équation de Schrödinger. On démontre la convergence L2 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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.07.018
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 = {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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[3] L.I. Ignat; E. Zuazua Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 7, pp. 529-534

[4] M. Keel; T. Tao Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980

[5] C.E. Kenig; G. Ponce; L. Vega Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991) no. 1, pp. 33-69

[6] M. Negreanu; E. Zuazua Convergence of a multigrid method for the controllability of a 1-d wave equation, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 338 (2004) no. 5, pp. 413-418

[7] A. Stefanov; P.G. Kevrekidis Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein–Gordon equations, Nonlinearity, Volume 18 (2005), pp. 1841-1857

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