Traveling Waves for Conservation Laws with Nonlocal Flux for Traffic Flow on Rough Roads

TL;DR

This paper analyzes traveling wave solutions for nonlocal traffic flow models on rough roads with discontinuous conditions, revealing conditions for existence, uniqueness, and stability of these profiles.

Contribution

It introduces and studies two new nonlocal conservation law models for traffic flow on rough roads, analyzing the existence and stability of traveling wave profiles.

Findings

Multiple profiles can exist depending on the case.

Some profiles are stable and serve as long-term solutions.

Profiles may be unique, multiple, or nonexistent based on conditions.

Abstract

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $x=0$. We study stationary traveling wave profiles cross $x=0$, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.