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On the continuum limit of the Follow-the-Leader model and its stability

Fabio Ancona, Mohamed Bentaibi, Francesco Rossi

Abstract

We consider the Follow-the-Leader (FtL) model and study which properties of the initial positioning of the vehicles ensure its convergence to the classical Lighthill-Whitham-Richards (LWR) model for traffic flow. Robustness properties of both FtL and LWR models with respect to the initial discretization schemes are investigated. Some numerical simulations are also discussed.

On the continuum limit of the Follow-the-Leader model and its stability

Abstract

We consider the Follow-the-Leader (FtL) model and study which properties of the initial positioning of the vehicles ensure its convergence to the classical Lighthill-Whitham-Richards (LWR) model for traffic flow. Robustness properties of both FtL and LWR models with respect to the initial discretization schemes are investigated. Some numerical simulations are also discussed.
Paper Structure (9 sections, 17 theorems, 153 equations, 5 figures)

This paper contains 9 sections, 17 theorems, 153 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Comparison with the literature
  3. The Follow-the-Leader model
  4. Eulerian and Lagrangian densities
  5. Cumulative and pseudo-inverse functions
  6. Evolution of the supports
  7. Convergence results for the cumulative and pseudo-inverse functions
  8. Proof of Theorem \ref{['thm:micromacrointro']}
  9. Proof of Theorem \ref{['thm:stabilitytheoremintro']}

Key Result

Theorem 1.1

Assume that the velocity map $v$ satisfies e-V1. Let $\bar{\rho} \in L^\infty(\mathbb{R}; [0,1])$ be with compact support and such that $\left\lVert\bar{\rho}\right\rVert_{L^1(\mathbb{R})}=1$. Let $\{x_j^N(t)\}_{j=0}^N$ be solutions of the FtL system FtL-0, FtL-0N, that moreover satisfy the conditio and that one of the two following conditions hold: Then the sequence ${\left\{\rho^{E,N}\right\}}_

Figures (5)

  • Figure 1: Problem statement
  • Figure 2: The Eulerian discrete density, the inverse Lagrangian discrete density and the (Dirac) empirical measure profiles ($N=4$).
  • Figure 3: Flux associated with \ref{['eq:v-bm']}.
  • Figure 4: Snapshots of the dynamics of $\rho^{E,5}(t)$ and $\tilde{\rho}^{E,5}(t)$.
  • Figure 5: Evolution of $\|\rho^{E,N}(t)-\tilde{\rho}^{E,N}(t)\|_{L^1(\mathbb{R})}$ for $N=5,20,100,500$.

Theorems & Definitions (51)

  • Definition 1.1: Condition of uniformly bounded initial support
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2: Discrete Eulerian Stability Theorem
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4: Discrete Minimum/Maximum Principle
  • Proposition 2.5
  • ...and 41 more