Comptes Rendus
no. 1-2
Partial Differential Equations
Stability results for the approximation of weakly coupled wave equations
[Résultats de stabilité de lʼapproximation de deux équations des ondes faiblement couplées]

Présenté par : Gilles Lebeau

Farah Abdallah1, 2 ; Serge Nicaise1 ; Julie Valein3 ; Ali Wehbe2
1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, institut des sciences et techniques, 59313 Valenciennes cedex 9, France
2 Université Libanaise, faculté des sciences 1 & Hadath, Beyrouth, Liban
3 Institut Elie-Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France
Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 29-34.

Dans cette Note, nous considérons lʼapproximation de deux équations des ondes couplées avec dissipation interne. Notre but est dʼamortir les modes étranges en introduisant des termes de viscosité numérique. Avec ces termes de viscosité, nous montrons la décroissance exponentielle ou polynomiale du schéma discret lorsque le problème continu a une telle décroissance (puisque le spectre de lʼopérateur spatial associé au système sans dissipation satisfait la condition du gap généralisé).

In this Note, we consider the approximation of two coupled wave equations with internal damping. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay (since the spectrum of the spatial operator associated with the undamped system satisfies the generalized gap condition).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.12.004
Farah Abdallah 1, 2 ; Serge Nicaise 1 ; Julie Valein 3 ; Ali Wehbe 2

1 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, institut des sciences et techniques, 59313 Valenciennes cedex 9, France
2 Université Libanaise, faculté des sciences 1 & Hadath, Beyrouth, Liban
3 Institut Elie-Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), B.P. 70239, 54506 Vandoeuvre-lès-Nancy cedex, France 
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     title = {Stability results for the approximation of weakly coupled wave equations},
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Farah Abdallah; Serge Nicaise; Julie Valein; Ali Wehbe. Stability results for the approximation of weakly coupled wave equations. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 29-34. doi : 10.1016/j.crma.2011.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.004/

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