Convergence order of upwind type schemes for transport equations with discontinuous coefficients

TL;DR

This paper analyzes the convergence order of upwind schemes for transport equations with discontinuous coefficients, establishing a convergence rate of 1/2 in Wasserstein distances and demonstrating its optimality.

Contribution

It provides the first rigorous proof of the convergence order 1/2 for upwind schemes under discontinuous coefficients and extends the result to other first-order schemes.

Findings

Convergence order of 1/2 in Wasserstein distance.

Optimality of the convergence order.

Applicability to other first-order schemes and semi-Lagrangian schemes.

Abstract

An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other "diffusive" "first order" schemes and to a forward semi-Lagrangian scheme.