Well-posedness and error estimates for coupled systems of nonlocal conservation laws

TL;DR

This paper establishes existence, uniqueness, and convergence rates for numerical solutions of coupled nonlocal hyperbolic conservation laws with rough fluxes, supported by numerical experiments.

Contribution

It introduces a framework for analyzing coupled nonlocal conservation laws with discontinuous fluxes, proving convergence rates and entropy solution existence.

Findings

Proved existence of entropy solutions with rough fluxes.

Derived convergence rates of 1/2 and 1/3 for finite volume methods.

Numerical experiments confirm theoretical convergence rates.

Abstract

This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: 1. Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform BV bound on the numerical approximations; 2. Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; 3. Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.