Probabilistic Analysis of the Upwind Scheme for Transport Equations |
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Authors: | Fran?ois?Delarue Email author" target="_blank">Frédéric?LagoutièreEmail author |
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Institution: | 1.Laboratoire Jean-Alexandre Dieudonné, CNRS UMR 6621,Université de Nice Sophia-Antipolis Parc Valrose,Nice Cedex 02,France;2.Département de Mathématiques, CNRS UMR 8628,Université Paris-Sud 11,Orsay Cedex,France;3.équipe-Projet SIMPAF,Centre de Recherche INRIA Futurs, Parc Scientifique de la Haute Borne,Villeneuve d’Ascq Cedex,France |
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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¥(0,T],L1(\mathbb Rd)){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¥(0,T] ×\mathbb Rd){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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