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

Fabio Ancona; Mohamed Bentaibi; Francesco Rossi

doi:10.3934/dcds.2024132

Article Contents

2025, Volume 45, Issue 5: 1366-1398. Doi: 10.3934/dcds.2024132

This issue Previous Article Next Article Existence of blow-up self-similar solutions for the supercritical quasilinear reaction-diffusion equation

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

1.
Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Italy
2.
Dipartimento di Culture del Progetto, Università Iuav di Venezia, Italy

^*Corresponding author: Francesco Rossi
^*Corresponding author: Francesco Rossi

Received: March 2024

Revised: September 2024

Early access: October 12, 2024

Published online: October 12, 2024

F. Ancona and F. Rossi are members of G.N.A.M.P.A. (I.N.d.A.M.).

Abstract / Introduction Full Text(HTML) Figure(5) Related Papers Cited by

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.

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Mathematics Subject Classification: Primary: 65M75; Secondary: 34D20, 35L65.

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Figure 1. Problem statement.

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Figure 2. The Eulerian discrete density, the inverse Lagrangian discrete density and the (Dirac) empirical measure profiles ($ N = 4 $).

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Figure 3. Flux associated with (83).

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Figure 4. Snapshots of the dynamics of $ \rho^{E, 5}(t) $ and $ \tilde \rho^{E, 5}(t) $.

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Figure 5. Evolution of $ \|\rho^{E, N}(t)-\tilde \rho^{E, N}(t)\|_{L^1( \mathbb{R})} $ for $ N = 5, 20, 100, 500 $.

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