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

Fundamental diagrams for kinetic equations of traffic flow

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.
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). 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/120897110. Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks - Conservation Laws Models,, AIMS Series on Applied Mathematics, (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. doi: 10.1137/S0036139902404700. Google Scholar

[13]

B. S. Kerner, The Physics of Traffic,, Springer, (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. 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. 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, (2009), 9727. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/120897110. Google Scholar

[11]

M. Garavello and B. Piccoli, Traffic Flow on Networks - Conservation Laws Models,, AIMS Series on Applied Mathematics, (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. doi: 10.1137/S0036139902404700. Google Scholar

[13]

B. S. Kerner, The Physics of Traffic,, Springer, (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. 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. 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, (2009), 9727. doi: 10.1007/978-0-387-30440-3_576. Google Scholar

[1]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[2]

Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic & Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033

[3]

Michael Herty, Lorenzo Pareschi, Mohammed Seaïd. Enskog-like discrete velocity models for vehicular traffic flow. Networks & Heterogeneous Media, 2007, 2 (3) : 481-496. doi: 10.3934/nhm.2007.2.481

[4]

Alexandre Bayen, Rinaldo M. Colombo, Paola Goatin, Benedetto Piccoli. Traffic modeling and management: Trends and perspectives. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : i-ii. doi: 10.3934/dcdss.2014.7.3i

[5]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255

[6]

Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773

[7]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287

[8]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[9]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161

[10]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic & Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[11]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060

[12]

Michael Herty, Reinhard Illner. Coupling of non-local driving behaviour with fundamental diagrams. Kinetic & Related Models, 2012, 5 (4) : 843-855. doi: 10.3934/krm.2012.5.843

[13]

Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311

[14]

Simone Göttlich, Oliver Kolb, Sebastian Kühn. Optimization for a special class of traffic flow models: Combinatorial and continuous approaches. Networks & Heterogeneous Media, 2014, 9 (2) : 315-334. doi: 10.3934/nhm.2014.9.315

[15]

Sharif E. Guseynov, Shirmail G. Bagirov. Distributed mathematical models of undetermined "without preference" motion of traffic flow. Conference Publications, 2011, 2011 (Special) : 589-600. doi: 10.3934/proc.2011.2011.589

[16]

Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks & Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015

[17]

Michael Burger, Simone Göttlich, Thomas Jung. Derivation of second order traffic flow models with time delays. Networks & Heterogeneous Media, 2019, 14 (2) : 265-288. doi: 10.3934/nhm.2019011

[18]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[19]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037

[20]

Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93-116. doi: 10.3934/krm.2017004

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]