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Probabilistic analysis of the upwind scheme for transport equations. (English) Zbl 1230.65008
The authors associate the upwind scheme for the \(d\)-dimensional transport problem \[ \begin{cases} \partial_{t}u(t,x)+\langle a(x),\nabla u(t,x)\rangle=0,\;(t,x)\in \mathbb{R}_{+}\times\mathbb{R}^{d},\\ u(0,x)=u^{0}(x),\;x\in\mathbb{R}^{d},\end{cases} \]
with a Markov chain. They show that the error induced by the scheme is governed by the fluctuations of the Markov chain around characteristics of the considered problem. Using the probabilistic approach, the authors obtain some results concerning the order of convergence of the proposed scheme.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
65C40 Numerical analysis or methods applied to Markov chains
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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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) · Zbl 1094.65089
[2] Bouchut F., Perthame B.: Kružkov’s estimates for scalar conservation laws revisited. Trans. Am. Math. Soc. 350(7), 2847–2870 (1998) · Zbl 0955.65069
[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) · Zbl 0921.65071
[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) · Zbl 0855.65103
[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) · Zbl 1127.65322
[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) · Zbl 0973.65078
[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 · Zbl 0981.65095
[8] Freedman D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975) · Zbl 0313.60037
[9] Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) · Zbl 0144.34903
[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) · Zbl 0734.60060
[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 · Zbl 0354.35021
[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) · Zbl 1171.35008
[14] Merlet B., Vovelle J.: Error estimate for finite volume scheme. Numer. Math. 106, 129–155 (2007) · Zbl 1116.35089
[15] Mizohata S.: The Theory of Partial Differential Equations. Cambridge University Press, New York (1973) · Zbl 0263.35001
[16] Norris J.R.: Markov Chains. Reprint of 1997 original. Cambridge University Press, Cambridge (1998)
[17] Petrov V.V.: Sums of Independent Random Variables. Springer-Verlag, New York (1975) · Zbl 0322.60043
[18] Ross S.M.: Introduction to Probability Models, 8th edn. Academic Press, Burlington (2003) · Zbl 1019.60003
[19] Shiryaev A.N.: Probability, 2nd edn. Springer-Verlag, New York (1996)
[20] Tang T., Teng Z.H.: The sharpness of Kuznetsov’s $${O(\(\backslash\)sqrt{\(\backslash\)Delta x}) L\(\backslash\)sp 1}$$ -error estimate for monotone difference schemes. Math. Comp. 64(210), 581–589 (1995) · Zbl 0845.65053
[21] Varadhan S.R.S.: Probability Theory. Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (2001) · Zbl 0980.60002
[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) · Zbl 0823.65087
[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) · Zbl 1030.65110
[24] Ziemer W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Springer, New York (1989) · Zbl 0692.46022
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