Probabilistic analysis of the upwind scheme for transport

TL;DR

This paper offers a probabilistic framework for analyzing the upwind scheme in multi-dimensional transport equations, revealing that errors are driven by Markov chain fluctuations and establishing convergence rates.

Contribution

It introduces a Markov chain-based probabilistic analysis of the upwind scheme, providing new insights into numerical diffusion and error behavior.

Findings

Fluctuations of the Markov chain are diffusive in nature.

The scheme achieves an order 1/2 convergence for BV initial data.

Convergence order is slightly less than 1/2 for Lipschitz initial data.

Abstract

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.