Probabilistic analysis of the upwind scheme for transport
For researchers in numerical analysis and PDEs, this work offers a novel probabilistic framework for understanding error in upwind schemes, though the results are incremental improvements over known convergence rates.
The paper provides a probabilistic analysis of the upwind scheme for transport equations, proving the scheme is of order 1/2 for BV initial data and order 1/2-a for Lipschitz continuous data, and offers a new interpretation of numerical diffusion.
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.