Deterministic particle approximation of the Hughes model in one space dimension
Marco Di Francesco Simone Fagioli Massimiliano Daniele Rosini Giovanni Russo
Kinetic & Related Models 2017, 10(1): 215-237 doi: 10.3934/krm.2017009

In this paper we present a new approach to the solution to a generalized version of Hughes' models for pedestrian movements based on a follow-the-leader many particle approximation. In particular, we provide a rigorous global existence result under a smallness assumption on the initial data ensuring that the trace of the solution along the turning curve is zero for all positive times. We also focus briefly on the approximation procedure for symmetric data and Riemann type data. Two different numerical approaches are adopted for the simulation of the model, namely the proposed particle method and a Godunov type scheme. Several numerical tests are presented, which are in agreement with the theoretical prediction.

keywords: Crowd dynamics conservation laws eikonal equation Hughes' model for pedestrian flows particle approximation
Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic
Marco Di Francesco Simone Fagioli Massimiliano D. Rosini
Mathematical Biosciences & Engineering 2017, 14(1): 127-141 doi: 10.3934/mbe.2017009

We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform ${\mathbf{BV}}$ estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

keywords: Aw-Rascle-Zhang model second order models for vehicular traffics many particle limit

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