Convergence rates for upwind schemes with rough coefficients

TL;DR

This paper analyzes the convergence rates of upwind finite volume schemes for continuity equations with rough velocity fields, establishing a weak convergence rate of at least 1/2 in mesh size, and demonstrating the optimality of this rate.

Contribution

It provides the first error estimates for upwind schemes with Sobolev regular velocity fields using Kantorovich-Rubinstein distance, revealing a weak convergence rate of 1/2.

Findings

Weak convergence rate of at least 1/2 in mesh size.

Optimality of the 1/2 rate demonstrated by example.

Error estimates derived using probabilistic interpretation.

Abstract

This paper is concerned with the numerical analysis of the explicit upwind finite volume scheme for numerically solving continuity equations. We are interested in the case where the advecting velocity field has spatial Sobolev regularity and initial data are merely integrable. We estimate the error between approximate solutions constructed by the upwind scheme and distributional solutions of the continuous problem in a Kantorovich-Rubinstein distance, which was recently used for stability estimates for the continuity equation by Seis [23]. Restricted to Cartesian meshes, our estimate shows that the rate of weak convergence is at least of order $1/2$ in the mesh size. The proof relies on a probabilistic interpretation of the upwind scheme Delarue and Lagouti\`ere [9]. We complement the weak convergence result with an example that illustrates that for rough initial data no rates can be expected in strong norms. The same example suggests that the weak order $1/2$ rate is optimal.