Issue |
ESAIM: Proc.
Volume 18, 2007
Paris-sud working group on modelling and scientific computing 2006-2007
|
|
---|---|---|
Page(s) | 38 - 47 | |
DOI | https://doi.org/10.1051/proc:071804 | |
Published online | 12 September 2007 |
On supra-convergence of the finite volume method for the linear advection problem
CMLA, ENS de Cachan, 61 Avenue du Président Wilson, 94235 Cachan
This paper investigates the supra-convergence phenomenon that one can observe in the upwind finite volume method for solving linear convection problem on a bounded domain but also in finite difference scheme with non-uniform grids. Although the scheme is no longer consistent in the finite difference sense and Lax-Richtmyer theorem not suitable, it is a well-known convergent method. In order to analyze the convergence rate, we introduce what we call a geometric corrector, which is associated with every finite volume mesh and every constant convection vector. Under a local quasi-uniformity condition and if the continuous solution is regular enough, there is a link between the convergence of the finite volume scheme and this geometric corrector : the study of this latter leads to the proof of the optimal order of convergence. We then focus our attention on an uniformly refined mesh of quadrangles and on a series of independent meshes of triangles and tetrahedrons. In these latter cases, a loss of accuracy is observed if there exists in the family of meshes a fixed straight line parallel to the convection direction.
Résumé
Nous nous intéressons au phénomène de supra-convergence que l'on observe pour le schéma volumes finis appliqué à l'équation d'advection linéaire dans un domaine borné mais aussi pour des schémas aux différences finies avec des maillages non uniformes. Bien que le schéma ne soit pas consistant au sens des différences finies et donc le théorème de Lax-Richtmyer non applicable, il est convergent. Pour mener à bien l'analyse, nous introduisons un correcteur géométrique ne dépendant que du maillage et du vecteur de convection. L'analyse de la convergence se ramène alors à l'analyse de ce correcteur si la solution est suffisamment régulière. Nous proposons l'étude numérique d'un maillage de quadrangles uniformément raffinés ainsi qu'une série de maillages de triangles et de tétraèdres indépendants les uns des autres. Dans ce dernier cas, on observe une perte dans le taux de convergence s'il persiste dans le maillage une droite fixe parallèle à la convection.
Mathematics Subject Classification: 65M06 / 65M12 / 65M15 / 65M50
Key words: Finite Volume Method ; Consistency and Accuracy ; Supra-Convergence ; linear convection problem
© EDP Sciences, ESAIM, 2007
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