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

Abstract

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. 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.