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

Abstract

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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.

How to cite

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Bouche, 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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  1. D. Bouche, J.-M. Ghidaglia and F. Pascal, Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation. SIAM J. Numer. Anal.43 (2005) 578–603.  Zbl1094.65089
  2. D. Bouche, J.-M. Ghidaglia and F. Pascal, An optimal a priori error analysis of the finite volume method for linear convection problems, in Finite volumes for complex applications IV, Problems and perspectives , F. Benkhaldoun, D. Ouazar and S. Raghay Eds., Hermes Science publishing, London, UK (2005) 225–236.  
  3. B. Cockburn, P.-A. Gremaud and J.X. Yang, A priori error estimates for numerical methods for scalar conservation laws. III: Multidimensional flux-splitting monotone schemes on non-cartesian grids. SIAM J. Numer. Anal.35 (1998) 1775–1803.  Zbl0909.65058
  4. L. Comtet, Advanced combinatorics – The art of finite and infinite expansions. D. Reidel Publishing Co., Dordrecht, The Netherlands (1974).  Zbl0283.05001
  5. F. Delarue and F. Lagoutière, Probabilistic analysis of the upwind scheme for transport equations. Arch. Ration. Mech. Anal. (to appear).  Zbl1230.65008
  6. B. Després, An explicit a priori estimate for a finite volume approximation of linear advection on non-cartesian grids. SIAM J. Numer. Anal.42 (2004) 484–504.  Zbl1127.65322
  7. B. Després, Lax theorem and finite volume schemes. Math. Comp.73 (2004) 1203–1234.  Zbl1053.65073
  8. G.P. Egorychev, Integral representation and the computation of combinatorial sums, Translations of Mathematical Monographs59. American Mathematical Society, Providence, USA (1984). [Translated from the Russian by H.H. McFadden, Translation edited by Lev J. Leifman.]  
  9. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis7, P.-A. Ciarlet and J.-L. Lions Eds., North-Holland (2000) 713–1020.  Zbl0981.65095
  10. W. Feller, An introduction to probability theory and its applicationsI. Third edition, John Wiley & Sons Inc., New York, USA (1968).  Zbl0155.23101
  11. S. Karlin, A first course in stochastic processes. Academic Press, New York, USA (1966).  
  12. D. Kröner, Numerical schemes for conservation laws. Wiley-Teubner Series Advances in Numerical Mathematics, Chichester: Wiley (1997).  Zbl0872.76001
  13. V. Lakshmikantham and D. Trigiante, Theory of difference equations: numerical methods and applications, 2nd edition, Monographs and Textbooks in Pure and Applied Mathematics251. Marcel Dekker Inc., New York, USA (2002).  Zbl1014.39001
  14. T.A. Manteuffel and A.B. White, Jr., The numerical solution of second order boundary value problems on nonuniform meshes. Math. Comput. 47 (1986) 511–535.  Zbl0635.65092
  15. B. Merlet, l∞ and l2 error estimate for a finite volume approximation of linear advection. SIAM J. Numer. Anal.46 (2009) 124–150.  Zbl1171.35008
  16. B. Merlet and J. Vovelle, Error estimate for the finite volume scheme applied to the advection equation. Numer. Math.106 (2007) 129–155.  Zbl1116.35089
  17. F. Pascal, On supra-convergence of the finite volume method. ESAIM: Proc.18 (2007) 38–47.  Zbl05213254
  18. T.E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal.28 (1991) 133–140.  Zbl0729.65085
  19. M. Renault, Lost (and found) in translation, André's actual method and its application to the generalized ballot problem. Amer. Math. Monthly115 (2008) 358–363.  Zbl1142.60004
  20. A. Tikhonov and A. Samarskij, Homogeneous difference schemes on non-uniform nets. U.S.S.R. Comput. Math. Math. Phys. 1963 (1964) 927–953.  Zbl0128.36702
  21. J.-P. Vila and P. Villedieu, Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math.94 (2003) 573–602.  Zbl1030.65110
  22. J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Numer. Math. 90 (2002) 563–596.  Zbl1007.65066
  23. B. Wendroff and A.B. White, Jr., Some supraconvergent schemes for hyperbolic equations on irregular grids, in Nonlinear hyperbolic equations – Theory, computation methods, and applications (Aachen, 1988), Notes Numer. Fluid Mech.24, Vieweg, Braunschweig, Germany (1989) 671–677.  
  24. B. Wendroff and A.B. White, Jr., A supraconvergent scheme for nonlinear hyperbolic systems. Comput. Math. Appl.18 (1989) 761–767.  Zbl0683.65078
  25. H.S. Wilf, generatingfunctionology. Third edition, A K Peters Ltd., Wellesley, USA (2006).  

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