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Comptes Rendus Mathématique
Volume 339, n° 10
pages 733-738 (novembre 2004)
Doi : 10.1016/j.crma.2004.09.037
Received : 4 June 2004 ;  accepted : 21 September 2004
Famille de schémas implicites uniformément contrôlables pour l'équation des ondes 1-D
Family of implicit schemes uniformly controllable for the 1-D wave equation.
 

Arnaud Münch
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Espagne 

Résumé

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 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. Pour citer cet article : A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.
Abstract

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  , 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. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.


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