On Traffic Flow with Nonlocal Flux: a Relaxation Representation

TL;DR

This paper models traffic flow with nonlocal flux using a relaxation approach, establishing uniform bounds and showing convergence to the entropy solution of a scalar conservation law as the nonlocal scale vanishes.

Contribution

It introduces a relaxation representation for nonlocal traffic flow models with exponential averaging kernels and proves convergence to the entropy solution.

Findings

Uniform BV bounds independent of the nonlocal scale

Convergence to the entropy solution as the scale parameter tends to zero

Uniqueness of the limit solution for affine velocity functions

Abstract

We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density $\rho$ ahead. The averaging kernel is of exponential type: $w_\varepsilon(s)=\varepsilon ^{-1} e^{-s/\varepsilon}$. By a transformation of coordinates, the problem can be reformulated as a $2\times 2$ hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $\varepsilon $. Letting $\varepsilon\to 0$, the limit yields a weak solution to the corresponding conservation law $\rho_t + ( \rho v(\rho))_x=0$. In the case where the velocity $v(\rho)= a-b\rho$ is affine, using the Hardy-Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.