# American Institute of Mathematical Sciences

February  2017, 14(1): 127-141. doi: 10.3934/mbe.2017009

## Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic

 1 DISIM, Università degli Studi dell'Aquila, via Vetoio 1 (Coppito), 67100 LAquila (AQ), Italy 2 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Sk lodowskiej 1, 20-031 Lublin, Poland

* Corresponding author: M. D. Rosini

Received  November 23, 2015 Accepted  April 15, 2016 Published  October 2016

Fund Project: MDF is supported by the Italian MIUR-PRIN project 2012L5W XHJ_003. SF is partially supported by the Italian INdAM-GNAMPA 2015 mini-project: Analisi e stabilità per modelli di equazioni alle derivate parziali nella matematica applicata. MDR is also partially supported by ICM Interdyscyplinarne Centrum Modelowania Matematycznego i Komputerowego, Uniwersytet Warszawski. The authors would like to thanks Giovanni Russo for comments and suggestions on the numerical part.

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.

Citation: Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini. Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences & Engineering, 2017, 14 (1) : 127-141. doi: 10.3934/mbe.2017009
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##### References:
Left column for Test 1 and right column for Test 2. Initial conditions are specified in the tables on the top using $N=200$ particles.
Left column for Test 3 and right column for Test 4. Initial conditions are specified in the tables on the top using $N=200$ particles.
Different numbers of particles and corresponding discrete $\mathbf{L^1}$-errors for densities.
 N Test 1 Test 2 Test 3 Test 4 100 8.9e − 03 4.1e − 03 4.7e − 03 2.1e − 03 500 1.8e − 03 1.1e − 03 1.8e − 03 4.7e − 04 1000 4.7e − 04 5.7e − 04 1.2e − 04 2.5e − 04 2000 4.5e − 04 3.4e − 04 8.2e − 04 1.3e − 04
 N Test 1 Test 2 Test 3 Test 4 100 8.9e − 03 4.1e − 03 4.7e − 03 2.1e − 03 500 1.8e − 03 1.1e − 03 1.8e − 03 4.7e − 04 1000 4.7e − 04 5.7e − 04 1.2e − 04 2.5e − 04 2000 4.5e − 04 3.4e − 04 8.2e − 04 1.3e − 04
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