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Issue ESAIM: M2AN
Volume 35, Number 6, November-December 2001
Page(s) 1159 - 1183
DOI 10.1051/m2an:2001152

DOI: 10.1051/m2an:2001152


M2AN, Vol. 35, N°6, pp. 1159-1183

Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Bruno Després1, 2 and Frédéric Lagoutière2

1  Commissariat à l'Énergie Atomique, BP 12, 91680 Bruyères-le-Châtel, France. (Bruno.Despres@cea.fr)
2  Laboratoire d'analyse numérique, Université de Paris VI, 175 rue du Chevaleret, 75013 Paris, France. (despres@ann.jussieu.fr)

(Received: April 26, 2001. Revised: July 19, 2001)

Abstract
We study a family of non linear schemes for the numerical solution of linear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate. This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.

Résumé
Nous étudions une famille de schémas non linéaires pour l'approximation numérique de l'advection linéaire sur grille quelconque en dimension d'espace supérieure à un. Une preuve de convergence est proposée à partir d'une estimation de la variation longitudinale. Cette estimation est une généralisation multidimensionnelle discrète de l'estimation TVD discrète, bien connue en dimension un d'espace.


AMS Subject: 76M12, 65M12

Key words: LVD estimate, Harten Formalism, linear advection, finite volume methods.


© EDP Sciences, SMAI 2001


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