Traveling Wave Profiles for a Follow-the-Leader Model for Traffic Flow with Rough Road Condition

TL;DR

This paper investigates a Follow-the-Leader traffic model with rough road conditions, deriving a DDDE for traveling wave profiles and analyzing their existence, uniqueness, and stability, offering a realistic alternative to classical methods.

Contribution

It introduces a novel DDDE framework for traveling wave profiles in FtL models with discontinuous road conditions, enhancing understanding of traffic flow dynamics.

Findings

Existence and uniqueness of traveling wave profiles established.

Profiles are locally stable under certain conditions.

Provides an alternative to vanishing viscosity methods.

Abstract

We study a Follow-the-Leader (FtL) ODE model for traffic flow with rough road condition, and analyze stationary traveling wave profiles where the solutions of the FtL model trace along, near the jump in the road condition. We derive a discontinuous delay differential equation (DDDE) for these profiles. For various cases, we obtain results on existence, uniqueness and local stability of the profiles. The results here offer an alternative approximation, possibly more realistic than the classical vanishing viscosity approach, to the conservation law with discontinuous flux for traffic flow.