On the accuracy of the finite volume approximations to nonlocal conservation laws

TL;DR

This paper establishes the first convergence rate for finite volume schemes approximating nonlocal scalar conservation laws, demonstrating a rate of .5 in L^1, with numerical experiments supporting the theoretical findings.

Contribution

It introduces a novel convergence proof and error analysis for finite volume schemes applied to nonlocal conservation laws without restrictive assumptions.

Findings

Finite volume schemes converge at a rate of .5 in L^1 norm.

First proof of convergence rate for this class of nonlocal conservation laws.

Numerical experiments confirm theoretical convergence rate.

Abstract

In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel $\mu$ or the flux $f$. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of $\sqrt{\Delta t}$ in $L^1(\mathbb{R})$. To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.