Probabilistic analysis of the upwind scheme for transport

Details Probabilistic analysis of the upwind scheme for transport Probabilistic analysis of the upwind scheme for transport

2007-12-19

arxiv.org/abs/0712.3217v1

Archive for Rational Mechanics and Analysis (2011) 229--268

Mathematics

Numerical Analysis

Scientific paper

We provide a probabilistic analysis of the upwind scheme for multi-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 prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon.

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