Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

Arnaud Münch

doi:10.1016/j.crma.2004.09.037

Analyse numérique

Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D

Présenté par : Olivier Pironneau

Arnaud Münch ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Espagne

Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 733-738.

Résumé
Abstract

Dans le cadre de la contrôlabilité exacte, on présente dans cette Note une famille paramétrée de schémas aux différences finis de l'équation des ondes 1-D. Ces schémas diffèrent des schémas centrés usuels par l'ajout de termes d'ordre $h^{2}$ , h désignant le pas de la discrétisation en espace. En s'appuyant sur une version discrète de l'inégalité d'Ingham pour les séries de Fourier non harmoniques, sur les propriétés spectrales et sur les diagrammes de dispersion des schémas, on détermine les paramètres assurant une propriété de contrôlabilité discrète uniforme en h et pour lesquels la condition de stabilité CFL est optimale.

We present a parameterized family of finite difference schemes for the exact controllability of the 1-D wave equation. These schemes differ from the usual centered ones by additional terms of order to $h^{2}$ , where h denotes the discretization step in space. Using a discrete version of Ingham's inequality for nonharmonic Fourier series, the spectral properties and dispersion diagrams of the schemes, we determine the parameters leading to a uniform controllability property with respect to h and an optimal stability CFL condition.

Reçu le : 2004-06-04
Accepté le : 2004-09-21
Publié le : 2004-11-05

DOI : 10.1016/j.crma.2004.09.037

Affiliations des auteurs :

Arnaud Münch ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Espagne

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     title = {Famille de sch\'emas implicites uniform\'ement contr\^olables pour l'\'equation des ondes {1-D}},
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Arnaud Münch. Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D. Comptes Rendus. Mathématique, Volume 339 (2004) no. 10, pp. 733-738. doi : 10.1016/j.crma.2004.09.037. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.037/

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Cité par 12 documents. Sources : Crossref, zbMATH

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