On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models

TL;DR

This paper proves the convergence of nonlocal balance laws to local entropy solutions in the context of nonlocal lane-changing traffic flow models, supported by numerical illustrations.

Contribution

It introduces a convergence result for nonlocal to local behavior in coupled balance laws with a focus on traffic flow applications.

Findings

Nonlocal operator converges to a Dirac distribution.

System converges to local entropy solutions.

Numerical simulations support theoretical results.

Abstract

We present a convergence result from nonlocal to local behavior for a system of nonlocal balance laws. The velocity field of the underlying conservation laws is diagonal. In contrast, the coupling to the remaining balance laws involves a nonlinear right-hand side that depends on the solution, nonlocal term, and other factors. The nonlocal operator integrates the density around a specific spatial point, which introduces nonlocality into the problem. Inspired by multi-lane traffic flow modeling and lane-changing, the nonlocal kernel is discontinuous and only looks downstream. In this paper, we prove the convergence of the system to the local entropy solutions when the nonlocal operator (chosen to be of an exponential type for simplicity) converges to a Dirac distribution. Numerical illustrations that support the main results are also presented.