Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux

TL;DR

This paper establishes the well-posedness and convergence of numerical schemes for a broad class of nonlinear nonlocal conservation laws with discontinuous flux, relevant to crowd dynamics and traffic flow, and connects nonlocal and local theories.

Contribution

It proves strong compactness, existence, and uniqueness of entropy solutions for nonlocal conservation laws with rough flux, a first in the literature, and demonstrates convergence of numerical schemes.

Findings

Strong compactness of Godunov and Lax-Friedrichs approximations

Existence and uniqueness of entropy solutions

Numerical experiments illustrating scheme performance

Abstract

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow, without any additional conditions on finiteness/discreteness of the set of discontinuities or on the monotonicity of the kernel/the discontinuous coefficient. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.