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Probabilistic Analysis of the Upwind Scheme for Transport Equations

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Abstract

We provide a probabilistic analysis of the upwind scheme for d-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we recover recent results due to Merlet and Vovelle (Numer Math 106: 129–155, 2007) and Merlet (SIAM J Numer Anal 46(1):124–150, 2007): we prove that the scheme is of order 1/2 in \({L^{\infty}([0,T],L^1(\mathbb R^d))}\) for an integrable initial datum of bounded variation and of order 1/2−ε, for all ε > 0, in \({L^{\infty}([0,T] \times \mathbb R^d)}\) for an initial datum of Lipschitz regularity. Our analysis provides a new interpretation of the numerical diffusion phenomenon.

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References

  1. Bouche D., Ghidaglia J.-M., Pascal F.: Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation. SIAM J. Numer. Anal. 43(2), 578–603 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchut F., Perthame B.: Kružkov’s estimates for scalar conservation laws revisited. Trans. Am. Math. Soc. 350(7), 2847–2870 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chainais-Hillairet C.: Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. M 2(N Math. Model. Numer. Anal. 33(1), 129–156 (1999)

    MathSciNet  Google Scholar 

  4. Cockburn B., Coquel F., Le Floch Ph.: An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63(207), 77–103 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Després B.: An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42(2), 484–504 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Eymard R., Gallouët T., Ghilani M., Herbin R.: Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volumes schemes. IMA J. Numer. Anal. 18, 563–594 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eymard, R., Gallouët, T., Herbin, R.: Finite Volume Method. Handbook for Numerical Analysis, Vol. VII (Eds. P. Ciarlet and J.-L. Lions). North Holland, Amsterdam, 2000

  8. Freedman D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975)

    Article  MATH  Google Scholar 

  9. Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  10. Godounov, S., Zabrodine, A., Ivanov, M., Kraĭko, A., Prokopov, G.: Résolution numérique des problèmes multidimensionnels de la dynamique des gaz. (French.) [Numerical solution of multidimensional problems of gas dynamics] Translated from the Russian by Valéri Platonov. Mir, Moscow, 1979

  11. Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer-Verlag, New York (1991)

    Book  MATH  Google Scholar 

  12. Kuznetsov N.N.: The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. Ž. Vyčisl. Mat. i Mat. Fiz. 16(6), 1489–1502 (1976) 1627

    MATH  Google Scholar 

  13. Merlet B.: L ∞- and L 2-error estimates for a finite volume approximation of linear advection. SIAM J. Numer. Anal. 46(1), 124–150 (2007)

    Article  MathSciNet  Google Scholar 

  14. Merlet B., Vovelle J.: Error estimate for finite volume scheme. Numer. Math. 106, 129–155 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mizohata S.: The Theory of Partial Differential Equations. Cambridge University Press, New York (1973)

    MATH  Google Scholar 

  16. Norris J.R.: Markov Chains. Reprint of 1997 original. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  17. Petrov V.V.: Sums of Independent Random Variables. Springer-Verlag, New York (1975)

    Google Scholar 

  18. Ross S.M.: Introduction to Probability Models, 8th edn. Academic Press, Burlington (2003)

    MATH  Google Scholar 

  19. Shiryaev A.N.: Probability, 2nd edn. Springer-Verlag, New York (1996)

    Google Scholar 

  20. Tang T., Teng Z.H.: The sharpness of Kuznetsov’s \({O(\sqrt{\Delta x}) L\sp 1}\)-error estimate for monotone difference schemes. Math. Comp. 64(210), 581–589 (1995)

    MATH  MathSciNet  ADS  Google Scholar 

  21. Varadhan S.R.S.: Probability Theory. Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (2001)

    Google Scholar 

  22. Vila J.-P.: Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO Modél. Math. Anal. Numér. 28(3), 267–295 (1994)

    MATH  MathSciNet  Google Scholar 

  23. Vila J.-P., Villedieu P.: Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math. 94(3), 573–602 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ziemer W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer, New York (1989)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jean-Alexandre Dieudonné, CNRS UMR 6621, Université de Nice Sophia-Antipolis Parc Valrose, 06108, Nice Cedex 02, France

    François Delarue

  2. Département de Mathématiques, CNRS UMR 8628, Université Paris-Sud 11, Bâtiment 425, 91405, Orsay Cedex, France

    Frédéric Lagoutière

  3. Équipe-Projet SIMPAF, Centre de Recherche INRIA Futurs, Parc Scientifique de la Haute Borne, 40, Avenue Halley, B.P. 70478, 59658, Villeneuve d’Ascq Cedex, France

    Frédéric Lagoutière

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  1. François Delarue
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Correspondence to Frédéric Lagoutière.

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Communicated by Y. Brenier

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Delarue, F., Lagoutière, F. Probabilistic Analysis of the Upwind Scheme for Transport Equations. Arch Rational Mech Anal 199, 229–268 (2011). https://doi.org/10.1007/s00205-010-0322-x

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