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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

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