Probabilistic Analysis of the Upwind Scheme for Transport Equations

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 R^d)){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 R^d){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.