[Un schéma de discrétisation bi-maille pour les équations de Schrödinger non-linéaires : propriétés dispersives et convergence]
On introduit une méthode bi-maille semi-discrète en différences finies pour l'approximation numérique de l'équation 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 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.
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@article{CRMATH_2005__341_6_381_0,
author = {Ignat, Liviu I. and Zuazua, Enrique},
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},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.07.018/}
}
TY - JOUR AU - Ignat, Liviu I. AU - Zuazua, Enrique TI - A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence JO - Comptes Rendus. Mathématique PY - 2005 SP - 381 EP - 386 VL - 341 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.07.018/ DO - 10.1016/j.crma.2005.07.018 LA - en ID - CRMATH_2005__341_6_381_0 ER -
%0 Journal Article %A Ignat, Liviu I. %A Zuazua, Enrique %T A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence %J Comptes Rendus. Mathématique %D 2005 %P 381-386 %V 341 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2005.07.018/ %R 10.1016/j.crma.2005.07.018 %G en %F CRMATH_2005__341_6_381_0
Ignat, Liviu I.; Zuazua, Enrique. A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence. Comptes Rendus. Mathématique, Tome 341 (2005) no. 6, pp. 381-386. doi : 10.1016/j.crma.2005.07.018. http://archive.numdam.org/articles/10.1016/j.crma.2005.07.018/
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