American Institute of Mathematical Sciences

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June  2014, 7(3): 449-462. doi: 10.3934/dcdss.2014.7.449

Fundamental diagrams for kinetic equations of traffic flow

Luisa Fermo 1, and  Andrea Tosin 2,

1. 

Department of Mathematics and Computer Science, University of Cagliari, Viale Merello 92, 09123 Cagliari, Italy
2. 

Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma

Received  May 2013 Revised  August 2013 Published  January 2014

In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called ``spatially homogeneous problem'' and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microscopic states. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.
Keywords: discrete kinetic models, Traffic flow, stochastic games, fundamental diagrams., asymptotic trends.
Mathematics Subject Classification: Primary: 90B20; Secondary: 34A34, 34D0.
Citation: Luisa Fermo, Andrea Tosin. Fundamental diagrams for kinetic equations of traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 449-462. doi: 10.3934/dcdss.2014.7.449
References:
[1]

A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), 1140003, 35 pp. doi: 10.1142/S0218202511400033.

[2]

S. Blandin, G. Bretti, C. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), 441-454. doi: 10.1016/j.amc.2009.01.057.

[3]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467.

[4]

I. Bonzani and L. Mussone, From experiments to hydrodynamic traffic flow models. I. Modelling and parameter identification, Math. Comput. Modelling, 37 (2003), 1435-1442. doi: 10.1016/S0895-7177(03)90051-3.

[5]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.

[6]

V. Coscia, M. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. II. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.

[7]

C. F. Daganzo, A variational formulation of kinematic waves: Solution methods, Transport. Res. B-Meth., 39 (2005), 934-950. doi: 10.1016/j.trb.2004.05.003.

[8]

P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279.

[9]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932. doi: 10.1142/S0218202507002157.

[10]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556. doi: 10.1137/120897110.

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks - Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[12]

M. Günther, A. Klar, T. Materne and R. Wegener, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2003), 468-483. doi: 10.1137/S0036139902404700.

[13]

B. S. Kerner, The Physics of Traffic, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.

[14]

J. Li and M. Zhang, Fundamental diagram of traffic flow, Transp. Res. Record, 2260 (2011), 50-59. doi: 10.3141/2260-06.

[15]

G. F. Newell, A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks, Transport. Res. B-Meth., 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D.

[16]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science, (ed., R. A. Meyers), 22, Springer, New York, 2009, 9727-9749. doi: 10.1007/978-0-387-30440-3_576.

show all references

References:
[1]

A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), 1140003, 35 pp. doi: 10.1142/S0218202511400033.

[2]

S. Blandin, G. Bretti, C. Cutolo and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), 441-454. doi: 10.1016/j.amc.2009.01.057.

[3]

S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127. doi: 10.1137/090754467.

[4]

I. Bonzani and L. Mussone, From experiments to hydrodynamic traffic flow models. I. Modelling and parameter identification, Math. Comput. Modelling, 37 (2003), 1435-1442. doi: 10.1016/S0895-7177(03)90051-3.

[5]

R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721. doi: 10.1137/S0036139901393184.

[6]

V. Coscia, M. Delitala and P. Frasca, On the mathematical theory of vehicular traffic flow. II. Discrete velocity kinetic models, Internat. J. Non-Linear Mech., 42 (2007), 411-421. doi: 10.1016/j.ijnonlinmec.2006.02.008.

[7]

C. F. Daganzo, A variational formulation of kinematic waves: Solution methods, Transport. Res. B-Meth., 39 (2005), 934-950. doi: 10.1016/j.trb.2004.05.003.

[8]

P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008), 279-293. doi: 10.3934/krm.2008.1.279.

[9]

M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), 901-932. doi: 10.1142/S0218202507002157.

[10]

L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to modeling vehicular traffic, SIAM J. Appl. Math., 73 (2013), 1533-1556. doi: 10.1137/120897110.

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks - Conservation Laws Models, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[12]

M. Günther, A. Klar, T. Materne and R. Wegener, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2003), 468-483. doi: 10.1137/S0036139902404700.

[13]

B. S. Kerner, The Physics of Traffic, Springer, Berlin, 2004. doi: 10.1007/978-3-540-40986-1.

[14]

J. Li and M. Zhang, Fundamental diagram of traffic flow, Transp. Res. Record, 2260 (2011), 50-59. doi: 10.3141/2260-06.

[15]

G. F. Newell, A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks, Transport. Res. B-Meth., 27 (1993), 289-303. doi: 10.1016/0191-2615(93)90039-D.

[16]

B. Piccoli and A. Tosin, Vehicular traffic: A review of continuum mathematical models, in Encyclopedia of Complexity and Systems Science, (ed., R. A. Meyers), 22, Springer, New York, 2009, 9727-9749. doi: 10.1007/978-0-387-30440-3_576.

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