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

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

Marco Di Francesco 1, , Simone Fagioli 1, and  Massimiliano D. Rosini 2,,

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.

Keywords: Aw-Rascle-Zhang model, second order models for vehicular traffics, many particle limit.
Mathematics Subject Classification: Primary: 35L65, 90B20.
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
References:
[1]

B. Andreianov, C. Donadello, U. Razafison, J. 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. Andreianov, C. 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. Aw, A. Klar, T. 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.

[4]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[5]

P. Bagnerini and M. Rascle, A multi-class homogenized hyperbolic model of traffic flow, SIAM Journal of Mathematical Analysis, 35 (2003), 949-973.  doi: 10.1137/S0036141002411490.

[6]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem vol. 20, Oxford university press, 2000.

[8]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.

[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.

[12]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, preprint.

[13]

R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci., 43 (2010), 203-223. 

[14]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear Follow-the-Leader models of traffic flow, Operations Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.

[15]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63.  doi: 10.1142/S0219891608001428.

[16]

M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[17]

S. Moutari and M. Rascle, A hybrid lagrangian model based on the Aw-Rascle traffic flow model, SIAM Journal on Applied Mathematics, 68 (2007), 413-436.  doi: 10.1137/060678415.

[18]

I. Prigogine and R. Herman, Kinetic theory of vehicular traffic IEEE Transactions on Systems, Man, and Cybernetics, 2 (1972), p295. doi: 10.1109/TSMC.1972.4309114.

[19]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[20]

A. I. Vol'pert, The spaces BV and quasilinear equations, (Russian) Mat. Sb. (N.S.), 73 (1967), 255-302. 

[21]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.

show all references

References:
[1]

B. Andreianov, C. Donadello, U. Razafison, J. 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. Andreianov, C. 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. Aw, A. Klar, T. 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.

[4]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.

[5]

P. Bagnerini and M. Rascle, A multi-class homogenized hyperbolic model of traffic flow, SIAM Journal of Mathematical Analysis, 35 (2003), 949-973.  doi: 10.1137/S0036141002411490.

[6]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220.  doi: 10.1007/s00205-007-0061-9.

[7]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem vol. 20, Oxford university press, 2000.

[8]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.  doi: 10.4310/CMS.2007.v5.n3.a2.

[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.

[12]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, preprint.

[13]

R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci., 43 (2010), 203-223. 

[14]

D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear Follow-the-Leader models of traffic flow, Operations Res., 9 (1961), 545-567.  doi: 10.1287/opre.9.4.545.

[15]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63.  doi: 10.1142/S0219891608001428.

[16]

M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.

[17]

S. Moutari and M. Rascle, A hybrid lagrangian model based on the Aw-Rascle traffic flow model, SIAM Journal on Applied Mathematics, 68 (2007), 413-436.  doi: 10.1137/060678415.

[18]

I. Prigogine and R. Herman, Kinetic theory of vehicular traffic IEEE Transactions on Systems, Man, and Cybernetics, 2 (1972), p295. doi: 10.1109/TSMC.1972.4309114.

[19]

P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.

[20]

A. I. Vol'pert, The spaces BV and quasilinear equations, (Russian) Mat. Sb. (N.S.), 73 (1967), 255-302. 

[21]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.

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.
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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.
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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
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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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