Geodesic
On supra-convergence of the finite volume method for the linear advection problem
Frédéric Pascal1
1 CMLA, ENS de Cachan, 61 Avenue du Président Wilson, 94235 Cachan
ESAIM. Proceedings, Tome 18 (2007), pp. 38-47.

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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.
DOI : 10.1051/proc:071804

Frédéric Pascal 1

1 CMLA, ENS de Cachan, 61 Avenue du Président Wilson, 94235 Cachan 
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Frédéric Pascal. On supra-convergence of the finite volume method for the linear advection problem. ESAIM. Proceedings, Tome 18 (2007), pp. 38-47. doi : 10.1051/proc:071804. http://geodesic.mathdoc.fr/articles/10.1051/proc:071804/

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