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
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