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

  • M. Di Francesco
  • S. Fagioli
  • M. D. Rosini
  • G. Russo
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


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.


Riemann Problem Entropy Solution Particle Method Vehicular Traffic Particle Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • M. Di Francesco
    • 1
  • S. Fagioli
    • 1
  • M. D. Rosini
    • 2
  • G. Russo
    • 3
  1. 1.DISIM, Università degli Studi dell’AquilaL’AquilaItaly
  2. 2.Instytut MatematykiUniwersytet Marii Curie-SkłodowskiejLublinPoland
  3. 3.Dipartimento di Matematica ed InformaticaUniversità di CataniaCataniaItaly

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