Relaxation and stability analysis of a third-order multiclass traffic flow model

2025/07/05 by Stephan Gerster, Giuseppe Visconti, Gerster, Stephan +1
Engineering · Social Sciences · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Optimization and Control (math.OC) #Traffic Prediction and Management Techniques #Traffic control and management #Transportation Planning and Optimization

paper · pdf · doi:10.48550/arxiv.2507.04135

openalex publication_date 2025/07/05 · openalex created_date 2025/10/20 · openalex updated_date 2026/07/28

Abstract

Traffic flow modeling spans a wide range of mathematical approaches, from microscopic descriptions of individual vehicle dynamics to macroscopic models based on aggregate quantities. A fundamental challenge in macroscopic modeling lies in the closure relations, particularly in the specification of a traffic hesitation function in second-order models like Aw-Rascle-Zhang. In this work, we propose a third-order hyperbolic traffic model in which the hesitation evolves as a driver-dependent dynamic quantity. Starting from a microscopic formulation, we relax the standard assumption by introducing an evolution law for the hesitation. This extension allows to incorporate hysteresis effects, modeling the fact that drivers respond differently when accelerating or decelerating, even under identical local traffic conditions. Furthermore, various relaxation terms are introduced. These allow us to establish relations to the Aw-Rascle-Zhang model and other traffic flow models.

Citations

• The hysteretic Aw–Rascle–Zhang model
• Limitations and Improvements of the Intelligent Driver Model (IDM)
• From heterogeneous microscopic traffic flow models to macroscopic models
• The BGK approximation of kinetic models for traffic
• A nonlinear discrete-velocity relaxation model for traffic flow
• Macroscopic modeling of multi-lane motorways using a two-dimensional\n second-order model of traffic flow
• Follow-the-Leader models can be viewed as a numerical approximation to\n the Lighthill-Whitham-Richards model for traffic flow
• Convergence of approximate solutions of conservation laws
• On the kinetic theory of vehicular traffic flow: Chapman-Enskog expansion versus Grad's moment method
• A Model for the Formation and Evolution of Traffic Jams
• Dynamical model of traffic congestion and numerical simulation

Related

• On the fundamental diagram of traffic flow
• A nonlinear discrete-velocity relaxation model for traffic flow
• The hysteretic Aw–Rascle–Zhang model
• A model for traffic flow encoding hysteresis
• Dynamical phases and hysteresis in a simple one-lane traffic model