We provide a probabilistic analysis of the upwind scheme formulti-dimensional transport equations. We associate a Markov chain with thenumerical scheme and then obtain a backward representation formula ofKolmogorov type for the numerical solution. We then understand that the errorinduced by the scheme is governed by the fluctuations of the Markov chainaround the characteristics of the flow. We show, in various situations, thatthe fluctuations are of diffusive type. As a by-product, we prove that thescheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for alla>0, for a Lipschitz continuous initial datum. Our analysis provides a newinterpretation of the numerical diffusion phenomenon.
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