Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson

¹ CEA, DAM, DIF, 91297 Arpajon, France.
² CMLA, ENS Cachan, CNRS, UniverSud, 61 avenue du Président Wilson, 94235 Cachan Cedex, France. frederic.pascal@cmla.ens-cachan.fr

Received: 8 July 2009
Revised: 17 December 2009

Abstract

When applied to the linear advection problem in dimension two, the upwind finite volume method is a non consistent scheme in the finite differences sense but a convergent scheme. According to our previous paper [Bouche et al., SIAM J. Numer. Anal. 43 (2005) 578–603], a sufficient condition in order to complete the mathematical analysis of the finite volume scheme consists in obtaining an estimation of order p, less or equal to one, of a quantity that depends only on the mesh and on the advection velocity and that we called geometric corrector. In [Bouche et al., Hermes Science publishing, London, UK (2005) 225–236], we prove that, on the mesh given by Peterson [SIAM J. Numer. Anal. 28 (1991) 133–140] and for a subtle alignment of the direction of transport parallel to the vertical boundary, the infinite norm of the geometric corrector only behaves like h_1/2 where h is a characteristic size of the mesh. This paper focuses on the case of an oblique incidence i.e. a transport direction that is not parallel to the boundary, still with the Peterson mesh. Using various mathematical technics, we explicitly compute an upper bound of the geometric corrector and we provide a probabilistic interpretation in terms of Markov processes. This bound is proved to behave like h, so that the order of convergence is one. Then the reduction of the order of convergence occurs only if the direction of advection is aligned with the boundary.