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

Daniel Bouche; Jean-Michel Ghidaglia; Frédéric P. Pascal

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 6, page 1279-1293
- ISSN: 0764-583X

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topBouche, Daniel, Ghidaglia, Jean-Michel, and Pascal, Frédéric P.. "Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1279-1293. <http://eudml.org/doc/250777>.

@article{Bouche2010,

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 h1/2
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.
},

author = {Bouche, Daniel, Ghidaglia, Jean-Michel, Pascal, Frédéric P.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite volume method; linear scalar problem; consistency
and accuracy; geometric corrector; advection problem with a constant velocity; theoretical analysis; counter-example of Peterson; upwind finite volume; two dimension},

language = {eng},

month = {10},

number = {6},

pages = {1279-1293},

publisher = {EDP Sciences},

title = {Theoretical analysis of the upwind finite volume scheme on the counter-example of Peterson},

url = {http://eudml.org/doc/250777},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Bouche, Daniel

AU - Ghidaglia, Jean-Michel

AU - Pascal, Frédéric P.

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/10//

PB - EDP Sciences

VL - 44

IS - 6

SP - 1279

EP - 1293

AB -
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 h1/2
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.

LA - eng

KW - Finite volume method; linear scalar problem; consistency
and accuracy; geometric corrector; advection problem with a constant velocity; theoretical analysis; counter-example of Peterson; upwind finite volume; two dimension

UR - http://eudml.org/doc/250777

ER -

## References

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