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

Stephan Gerster, Giuseppe Visconti

TL;DR

This work introduces a third-order hyperbolic traffic model (TOM) in which driver hesitation is a dynamic, hysteresis-enabled quantity rather than a fixed closure. The macroscopic TOM forms a Temple-class hyperbolic system, with a rigorous entropy structure, and reduces to the Aw-Rascle-Zhang (ARZ) model under relaxation scenarios. The authors analyze relaxation to ARZ and to multiclass velocity curves using Chapman-Enskog expansions, deriving conditions for dissipative diffusion and discussing sub-characteristic-type stability. They also examine the Riemann problem to illustrate how hysteresis alters shock/rarefaction structures and accelerator/decelerator responses, establishing TOM as a flexible framework linking microscopic interactions to macroscopic traffic behavior.

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.

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

TL;DR

This work introduces a third-order hyperbolic traffic model (TOM) in which driver hesitation is a dynamic, hysteresis-enabled quantity rather than a fixed closure. The macroscopic TOM forms a Temple-class hyperbolic system, with a rigorous entropy structure, and reduces to the Aw-Rascle-Zhang (ARZ) model under relaxation scenarios. The authors analyze relaxation to ARZ and to multiclass velocity curves using Chapman-Enskog expansions, deriving conditions for dissipative diffusion and discussing sub-characteristic-type stability. They also examine the Riemann problem to illustrate how hysteresis alters shock/rarefaction structures and accelerator/decelerator responses, establishing TOM as a flexible framework linking microscopic interactions to macroscopic traffic behavior.

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.

Paper Structure

This paper contains 13 sections, 6 theorems, 81 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Microscopic description of traffic flow
  3. Follow-the-Leader models
  4. Multiclass models
  5. The third-order model (TOM)
  6. The macroscopic TOM model
  7. Macroscopic traffic flow models with relaxation
  8. Relaxation in the Aw-Rascle-Zhang model
  9. Relaxation to the homogeneous Aw-Rascle-Zhang model
  10. Analysis of the frozen characteristics
  11. Relaxation to a prescribed velocity of a multiclass system
  12. Discussion of the Riemann problem
  13. Summary

Key Result

lemma 1

The formulations PiuS1 -- PiuS3 are equivalent for smooth solutions.

Figures (4)

  • Figure 1: Panel (i) -- (iii) state the solution to the Riemann problem at time $t=0.3$ for the choice $q(h)=h^{1/2}$. Panel (iv) highlights the difference $\Delta v(t,x)$ over a rarefaction wave.
  • Figure 2: Panel (i) -- (iii) state the solution to the Riemann problem at time $t=0.3$ for the choice $q(h)=h$, when the third-order model coincides with the Aw-Rascle-Zhang-model. Panel (iv) shows that the acceleration of drivers is constant over a rarefaction wave.
  • Figure 3: Panel (i) -- (iii) state the solution to the Riemann problem at time $t=0.3$ for the choice $q(h)=h^2$. Panel (iv) highlights the difference $\Delta v(t,x)$ over a rarefaction wave.
  • Figure 4: Solutions to a Riemann problem with a shock, corresponding to the first field, and a contact with speed $v_r$.

Theorems & Definitions (14)

  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • remark 2
  • ...and 4 more