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On a degenerate second order traffic model: existence of discrete evolutions, deterministic many-particle limit and first order approximation

Dario Mazzoleni, Emanuela Radici, Filippo Riva

TL;DR

The paper develops a degenerate microscopic second-order traffic model where the inertia term is modulated by a density-dependent factor, capturing increased driver attentiveness under congestion. It proves existence of solutions, derives a discrete-to-continuum limit to a degenerate pressureless Euler-type system, and analyzes a vanishing-inertia regime that justifies first-order macroscopic models with nonlinear mobility. A toy traffic-light example illustrates the model's qualitative behavior, while the analysis relies on piecewise-constant approximations to obtain rigorous weak solutions. Overall, the work connects a novel microscopic scheme to a macroscopic description, offering a rigorous basis for nonlinear mobility-based first-order traffic modeling and comparisons to ARZ.

Abstract

We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with the intent of considering the attentiveness of the drivers in dependence on the amount of congestion. Such term makes the system highly degenerate; indeed, coherently with the modellistic viewpoint, we allow for the nonlinearity to vanish as soon as consecutive vehicles are very close to each other. We first show existence of solutions to the degenerate discrete system. We then perform a rigorous discrete-to-continuum limit, as the number of vehicles grows larger and larger, by making use of suitable piece-wise constant approximations of the relevant macroscopic variables. The resulting continuum system turns out to be described by a degenerate pressure-less Euler-type equation, and we discuss how this could be considered an alternative to the groundbreaking Aw-Rascle-Zhang traffic model. Finally, we study the singular limit to first order dynamics in the spirit of a vanishing-inertia argument. This eventually validates the use of first order macroscopic models with nonlinear mobility to describe a congested traffic stream.

On a degenerate second order traffic model: existence of discrete evolutions, deterministic many-particle limit and first order approximation

TL;DR

The paper develops a degenerate microscopic second-order traffic model where the inertia term is modulated by a density-dependent factor, capturing increased driver attentiveness under congestion. It proves existence of solutions, derives a discrete-to-continuum limit to a degenerate pressureless Euler-type system, and analyzes a vanishing-inertia regime that justifies first-order macroscopic models with nonlinear mobility. A toy traffic-light example illustrates the model's qualitative behavior, while the analysis relies on piecewise-constant approximations to obtain rigorous weak solutions. Overall, the work connects a novel microscopic scheme to a macroscopic description, offering a rigorous basis for nonlinear mobility-based first-order traffic modeling and comparisons to ARZ.

Abstract

We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with the intent of considering the attentiveness of the drivers in dependence on the amount of congestion. Such term makes the system highly degenerate; indeed, coherently with the modellistic viewpoint, we allow for the nonlinearity to vanish as soon as consecutive vehicles are very close to each other. We first show existence of solutions to the degenerate discrete system. We then perform a rigorous discrete-to-continuum limit, as the number of vehicles grows larger and larger, by making use of suitable piece-wise constant approximations of the relevant macroscopic variables. The resulting continuum system turns out to be described by a degenerate pressure-less Euler-type equation, and we discuss how this could be considered an alternative to the groundbreaking Aw-Rascle-Zhang traffic model. Finally, we study the singular limit to first order dynamics in the spirit of a vanishing-inertia argument. This eventually validates the use of first order macroscopic models with nonlinear mobility to describe a congested traffic stream.
Paper Structure (17 sections, 13 theorems, 127 equations)

This paper contains 17 sections, 13 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Overview of the literature and of the existing models
  3. A new second order model
  4. Main results
  5. Comparison with the ARZ model
  6. Setting and main results
  7. Degenerate microscopic traffic model
  8. Discrete-to-continuum limit
  9. Asymptotics to first-order dynamics
  10. A toy example of a traffic light
  11. The vehicle in front
  12. The vehicle in between
  13. The vehicle behind
  14. Preliminary tools
  15. Existence of solutions to the discrete system
  16. ...and 2 more sections

Key Result

Theorem 2.4

Under the previous assumptions, there exists a solution $x$ to the microscopic traffic model eq:system in the sense of Definition def:sol satisfying the following bound on the velocity for $i=0,\ldots,N-1$: where we set and $\bar{\vartheta}:=\max\limits_{[0,\bar{\rho}_\vartheta]}\vartheta$. If in addition $x_N$ is strictly increasing and $F$ is positive, then $x_i$ is strictly increasing for all

Theorems & Definitions (36)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9: Partial uniqueness
  • Definition 2.10
  • ...and 26 more