The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance

TL;DR

This paper proves that the known first-order convergence rate of monotone schemes for conservation laws in the Wasserstein distance is optimal for certain initial data, and suggests it may be worse for unbounded data.

Contribution

It establishes the optimality of the first-order convergence rate for monotone schemes in the Wasserstein distance and explores the impact of initial data bounds.

Findings

First-order convergence rate is proven to be optimal for compactly supported, Lipschitz-bounded initial data.

Numerical evidence suggests the convergence rate deteriorates for unbounded initial data.

The results clarify the limitations of monotone schemes in Wasserstein distance approximation.

Abstract

In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone finite volume approximations of conservation laws. For compactly supported, $\Lip^+$-bounded initial data they showed a first-order convergence rate in the Wasserstein distance. Our main result is to prove that this rate is optimal. We further provide numerical evidence indicating that the rate in the case of $\Lip^+$-unbounded initial data is worse than first-order.