Free access article

ESAIM: Proc., 2005, Vol. 15, pp. 58-74
DOI: 10.1051/proc:2005021

Conservative and entropic numerical schemes for the non homogeneous Fokker-Planck-Landau equation

Nicolas Crouseilles

Mathématiques pour l'Industrie et la Physique, CNRS UMR 5640, INSA et Université Paul Sabatier - Toulouse 3, 118 route de Narbonne, F-31062 Toulouse cedex 4, France.

crouseilles@mip.ups-tlse.fr

Abstract
In this paper, we investigate the approximation of the solution to the Vlasov equation coupled with the Fokker-Planck-Landau collision operator using a phase space grid. On the one hand, the transport is treated by a flux conservative method and the distribution function is reconstructed to to preserve energy and to control numerical oscillations. On the other hand, the collision operators are approximated to preserve the main properties (conservations and decrease of the entropy). Several numerical results are presented in one dimension in space and three dimensions in velocity.

Résumé
Dans cet article, on étudie l'approximation de la solution de l'équation de Vlasov couplée avec l'opérateur de collisions de Fokker-Planck-Landau utilisant une grille de l'espace des phases. D'une part, le transport est traité par une méthode à flux conservatif et la fonction de distribution est reconstruite pour préserver l'énergie et contrôler les oscillations numériques. D'autre part, les opérateurs de collisions sont approchés de façon à préserver leurs propriétés fondamentales (lois de conservations et décroissance de l'entropie). Divers résultats numériques sont présentés en dimension un d'espace et dimension trois en vitesse.

Key words: Landau-Fokker-Planck equation, Vlasov-Poisson system, finite difference methods


© EDP Sciences, ESAIM 2005

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