Second-order convergence of monotone schemes for conservation laws

TL;DR

This paper proves that certain monotone schemes for scalar conservation laws achieve second-order convergence in Wasserstein distance under specific initial conditions, a first in the field for discontinuous solutions.

Contribution

It establishes second-order convergence of monotone, $W_1$-contractive schemes for scalar conservation laws, including Lax--Friedrichs, Enquist--Osher, and Godunov schemes, in Wasserstein distance.

Findings

W_1-contractive schemes converge at rate Δx^2

Lax--Friedrichs, Enquist--Osher, and Godunov schemes are W_1-contractive

Numerical experiments confirm theoretical results

Abstract

We prove that a class of monotone, \emph{$W_1$-contractive} schemes for scalar conservation laws converge at a rate of $\Delta x^2$ in the Wasserstein distance ($W_1$-distance), whenever the initial data is decreasing and consists of a finite number of piecewise constants. It is shown that the Lax--Friedrichs, Enquist--Osher and Godunov schemes are $W_1$-contractive. Numerical experiments are presented to illustrate the main result. To the best of our knowledge, this is the first proof of second-order convergence of any numerical method for discontinuous solutions of nonlinear conservation laws.