https://ntrs.nasa.gov/api/citations/20110011294/downloads/20110011294.pdf

Notes on accuracy of finite-volume discretization schemes on irregular grids

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

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.

/pdf/theoretical-analysis-of-the-upwind-finite-volume-scheme-on-33ec9dblo8.pdf

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

/pdf/unsteady-corrector-method-for-accuracy-analysis-of-linear-4feg1blgz1.pdf

Unsteady corrector method for accuracy analysis of linear numerical schemes for transport equation

/pdf/unsteady-geometric-corrector-and-the-error-estimate-of-2nmhv7lmb2.pdf

Unsteady geometric corrector and the error estimate of finite volume method on unstructured meshes

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. Resume. Nous nous interessons au phenomene de supra-convergence que l'on observe pour le schema volumes finis appliquel'´equation d'advection lineaire dans un domaine borne mais aussi pour des schemas aux differences finies avec des maillages non uniformes. Bien que le schema ne soit pas consistant au sens des differences finies et donc le theoreme de Lax-Richtmyer non applicable, il est convergent. Pour menerbien l'analyse, nous introduisons un correcteur geometrique ne dependant que du maillage et du vecteur de convection. L'analyse de la convergence se ramene alorsl'analyse de ce correcteur si la solution est suffisamment reguliere. Nousproposons l'´etude numerique d'un maillage de quadrangles uniformement raffines ainsi qu'une serie de maillages de triangles et de tetraedres independants 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 parallelela convection.

/pdf/on-supra-convergence-of-the-finite-volume-method-for-the-e4kqsib8dd.pdf

On supra-convergence of the finite volume method for the linear advection problem

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On supra-convergence of the finite volume method for the linear advection problem