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

  • * Corresponding author: M. D. Rosini

    * Corresponding author: M. D. Rosini
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.
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  • 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.

    Mathematics Subject Classification: Primary: 35L65, 90B20.


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  • Figure 1.  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.

    Figure 2.  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.

    Table 1.  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
     | Show Table
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  • [1] B. AndreianovC. DonadelloU. RazafisonJ. Y. Rolland and M. D. Rosini, Solutions of the Aw-Rascle-Zhang system with point constraints, Networks and Heterogeneous Media, 11 (2016), 29-47.  doi: 10.3934/nhm.2016.11.29.
    [2] B. AndreianovC. Donadello and M. D. Rosini, A second-order model for vehicular traffics with local point constraints on the flow, Mathematical Models and Methods in Applied Sciences, 26 (2016), 751-802.  doi: 10.1142/S0218202516500172.
    [3] A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic Follow-the-Leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.
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    [9] M. Di Francesco and M. Rosini, Rigorous derivation of nonlinear scalar conservation laws from Follow-the-Leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831-871.  doi: 10.1007/s00205-015-0843-4.
    [10] M. Di Francesco, S. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, preprint, arXiv: 1605.05883.
    [11] M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Deterministic particle approximation of the Hughes model in one space dimension, preprint, arXiv: 1602.06153.
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