Iterative Methods for Model Reduction by Domain Decomposition
نویسندگان
چکیده
We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the loworder model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including the Laplace equation with non-linear boundary conditions and the compressible Euler equations. Key-words: low-order models, domain decomposition, compressible flows ∗ INRIA Futurs, Equipe-Projet MC2 and Institut de Mathématiques de Bordeaux, UMR 5251 CNRS, Université Bordeaux 1, 33405 Talence cedex, France. † Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino. 10129 Torino, Italy Méthodes Itératives pour la Réduction de Modèles par Décomposition de Domaine Résumé : On propose une méthode pour réduire les efforts de calcul pour résoudre une équation aux dérivées partielles sur un domaine donné. L’idée principale est de diviser le domaine considéré en deux sous-domaines, et d’employer différentes méthodes d’approximation dans chacun des deux sous-domaines. En particulier, dans un des sous-domaines l’équation en question est discrétisée par une méthode canonique, tandis que dans l’autre un modèle d’ordre réduit du problème original est utilisé. Des stratégies différentes pour coupler le modèle d’ordre réduit á la discrétisation habituelle sont présentés. L’efficacité de ces approches est testée sur des exemples numériques pertinentes pour des problèmes modèles non linéaires, notamment l’équation de Laplace, avec des conditions limites non linéaires, et les équations d’Euler compressibles. Mots-clés : modèles réduits, décomposition de domaine, écoulements compressibles Iterative Methods for Model Reduction by Domain Decomposition 3
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