Influence of boundary layers over the rate of convergence in a penalization method for a 1-D wave equation

Bruno Fornet

doi:10.1016/j.crme.2008.01.009

Influence of boundary layers over the rate of convergence in a penalization method for a 1-D wave equation

Presented by: Évariste Sanchez-Palencia

Bruno Fornet ^1,²

¹ LATP, université de Provence, 39, rue Joliot-Curie, 13453 Marseille cedex 13, France
² LMRS, université de Rouen, avenue de l'université, B.P. 12, 76801 Saint Étienne du Rouvray, France

Comptes Rendus. Mécanique, Volume 336 (2008) no. 5, pp. 454-457.

Abstract
Résumé

For domain penalization methods, there can be a gap between the expected speed of convergence and the observed one, by numerical means. Such a gap has been observed by Paccou, et al. [A. Paccou, G. Chiavassa, J. Liandrat, K. Schneider, A penalization method applied to the wave equation, C. R. Mecanique 333 (2005) 79–85], concerning the penalization of a wave equation. We answer here one of their questions by proving that the observed lack in convergence speed is clearly related to the formation of boundary layers on one side of the boundary.

Pour les méthodes de pénalisation de domaine, il est possible qu'il y ait un écart entre la vitesse de convergence attendue et celle observée numériquement. Un tel écart a été mis en évidence par Paccou, et al. [A. Paccou, G. Chiavassa, J. Liandrat, K. Schneider, A penalization method applied to the wave equation, C. R. Mecanique 333 (2005) 79–85], lors de la pénalisation d'une équation des ondes. On répond ici à une question posée dans, en prouvant que le défaut de vitesse de convergence observé est clairement provoqué par la formation de couches limites, localisées d'un seul côté du bord.

Received: 2007-11-05
Accepted: 2008-01-29
Published online: 2008-03-03

DOI: 10.1016/j.crme.2008.01.009

Keywords: Computational fluid mechanics, Domain penalization method, Boundary layers, Wave equation
Mots-clés : Mécanique des fluides numérique, Méthode de pénalisation de domaine, Couches limites, Équation des ondes

Author's affiliations:

Bruno Fornet ^{1, 2}

¹ LATP, université de Provence, 39, rue Joliot-Curie, 13453 Marseille cedex 13, France
² LMRS, université de Rouen, avenue de l'université, B.P. 12, 76801 Saint Étienne du Rouvray, France

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     title = {Influence of boundary layers over the rate of convergence in a penalization method for a {1-D} wave equation},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {454--457},
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     year = {2008},
     doi = {10.1016/j.crme.2008.01.009},
     language = {en},
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Bruno Fornet. Influence of boundary layers over the rate of convergence in a penalization method for a 1-D wave equation. Comptes Rendus. Mécanique, Volume 336 (2008) no. 5, pp. 454-457. doi : 10.1016/j.crme.2008.01.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.01.009/

References
Cited by

[1] Ph. Angot; Ch.H. Bruneau; P. Fabrie A penalization method to take into account obstacles in viscous flows, Numer. Math., Volume 81 (1999), pp. 497-520

[2] B. Fornet, O. Guès, Penalization approach of semi-linear symmetric hyperbolic problems with dissipative boundary conditions, preprint available on HAL, 2007

[3] A. Paccou; G. Chiavassa; J. Liandrat; K. Schneider A penalization method applied to the wave equation, C. R. Mecanique, Volume 333 (2005), pp. 79-85

[4] C. Bardos; J. Rauch Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc., Volume 270 (1982), pp. 377-408

[5] J. Rauch, Boundary value problems as limits of problems in all space, Séminaire Goulaouic–Schwartz (1978/1979), exposé No. 3

[6] J. Droniou, Rapport de stage à l'université de Nice, unpublished, 1997

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