Asymptotic Preserving numerical schemes for multiscale parabolic problems

Crouseilles, Nicolas; Lemou, Mohammed; Vilmart, Gilles

doi:10.1016/j.crma.2015.11.010

Partial differential equations/Numerical analysis

Asymptotic Preserving numerical schemes for multiscale parabolic problems

Presented by: Olivier Pironneau

Crouseilles, Nicolas ^1,²; Lemou, Mohammed ^3,²; Vilmart, Gilles ⁴

¹ INRIA, France
² IRMAR, Université de Rennes-1, campus de Beaulieu, F-35042 Rennes Cedex, France
³ CNRS, France
⁴ Université de Genève, Section de mathématiques, 2–4, rue du Lièvre, CP 64, CH-1211 Genève 4, Switzerland

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 271-276.

Abstract
Résumé

We consider a class of multiscale parabolic problems with diffusion coefficients oscillating in space at a possibly small scale ε. Numerical homogenization methods are popular for such problems, because they capture efficiently the asymptotic behavior as $ε \to 0$ , without using a dramatically fine spatial discretization at the scale of the fast oscillations. However, it is known that such homogenization schemes are in general not accurate for both the highly oscillatory regime $ε \to 0$ and the non-oscillatory regime $ε \sim 1$ . In this paper, we introduce an Asymptotic Preserving method based on an exact micro–macro decomposition of the solution, which remains consistent for both regimes.

On considère une classe de problèmes paraboliques multi-échelles dont les coefficients de diffusion oscillent rapidement en espace à une échelle ε possiblement petite. Les méthodes numériques d'homogénéisation sont populaires pour ces problèmes, car elles capturent efficacement le comportement asymptotique lorsque $ε \to 0$ , sans utiliser une discrétisation spatiale aussi fine que l'échelle des oscillations rapides, comme le nécessiteraient les méthodes non raides standard. Cependant, les schémas d'homogénéisation existants ne sont en général pas précis dans les deux régimes oscillant $ε \to 0$ et non oscillant $ε \sim 1$ . Dans ce travail, nous introduisons une méthode Asymptotic Preserving basée sur une décomposition micro–macro exacte, qui reste consistante pour les deux régimes.

Received: 2015-07-27
Accepted: 2015-11-19
Published online: 2016-02-04

DOI: 10.1016/j.crma.2015.11.010

Author's affiliations:

Crouseilles, Nicolas ^{1, 2}; Lemou, Mohammed ^{3, 2}; Vilmart, Gilles ⁴

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     author = {Crouseilles, Nicolas and Lemou, Mohammed and Vilmart, Gilles},
     title = {Asymptotic {Preserving} numerical schemes for multiscale parabolic problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {271--276},
     publisher = {Elsevier},
     volume = {354},
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Crouseilles, Nicolas; Lemou, Mohammed; Vilmart, Gilles. Asymptotic Preserving numerical schemes for multiscale parabolic problems. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 271-276. doi : 10.1016/j.crma.2015.11.010. https://www.numdam.org/articles/10.1016/j.crma.2015.11.010/

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