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On application of optimal control to SEIR normalized models: Pros and cons
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |


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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