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Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows

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Active Particles, Volume 1

Abstract

We review the recent results and present new ones on a deterministic follow-the-leader particle approximation of first-and second-order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first-order Lighthill–Whitham–Richards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretized vehicles seen as moving particles. The convergence of the scheme in the many particle limit toward the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the initial–boundary value problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case, we prove that our scheme is convergent strongly in \(\mathbf {L^{1}}\) up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second-order Aw–Rascle–Zhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases.

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Acknowledgements

MDF and MDR are supported by the GNAMPA (Italian group of Analysis, Probability, and Applications) project Geometric and qualitative properties of solutions to elliptic and parabolic equations. SF and MDR are supported by the GNAMPA (Italian group of Analysis, Probability, and Applications) project Analisi e stabilità per modelli di equazioni alle derivate parziali nella matematica applicata. GR was partially supported by ITN-ETN Marie Curie Actions ModCompShock—‘Modeling and Computation of Shocks and Interfaces.’

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Authors and Affiliations

  1. DISIM, Università degli Studi dell’Aquila, via Vetoio 1 (Coppito), 67100, L’Aquila, AQ, Italy

    M. Di Francesco & S. Fagioli

  2. Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, plac Marii Curie-Skłodowskiej 1, 20-031, Lublin, Poland

    M. D. Rosini

  3. Dipartimento di Matematica ed Informatica, Università di Catania, Viale Andrea Doria 6, 95125, Catania, Italy

    G. Russo

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  1. M. Di Francesco
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Correspondence to M. Di Francesco .

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Editors and Affiliations

  1. Department of Mathematical Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Pierre Degond

  3. CSCAMM, University of Maryland, College Park, USA

    Eitan Tadmor

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Di Francesco, M., Fagioli, S., Rosini, M.D., Russo, G. (2017). Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-49996-3_9

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