Vehicular Traffic: A Review of Continuum Mathematical Models

This paper reviews some of the major macroscopic and kinetic mathematical models of vehicular traffic available in the specialized literature. Application to road networks is also discussed, and an extensive list of references is provided to both the original works presented here and other sources for further details. Microscopic models are only briefly recalled in the Introduction for the sake of thoroughness, as they are actually beyond the scope of the review. However, suitable references are included for the interested readers

[1]  Salissou Moutari,et al.  Multi-class traffic models on road networks , 2006 .

[2]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[3]  E. Angelis.  Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems , 1999 .

[4]  Harold J Payne,et al.  MODELS OF FREEWAY TRAFFIC AND CONTROL. , 1971 .

[5]  Ciro D'Apice,et al.  VERTEX FLOW MODELS FOR VEHICULAR TRAFFIC ON NETWORKS , 2008 .

[6]  Nicola Bellomo,et al.  The Enskog equation , 1991 .

[7]  C. Daganzo.  Requiem for second-order fluid approximations of traffic flow , 1995 .

[8]  Andrea Tosin,et al.  Mathematical modeling of vehicular traffic: a discrete kinetic theory approach , 2007 .

[9]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[10]  Paolo Frasca,et al.  On the mathematical theory of vehicular traffic flow II: Discrete velocity kinetic models , 2007 .

[11]  Benedetto Piccoli,et al.  Traffic circles and timing of traffic lights for cars flow , 2005 .

[12]  Mauro Garavello,et al.  A Well Posed Riemann Problem for the p-System at a Junction , 2006, Networks Heterog. Media.

[13]  Kai Nagel,et al.  Still Flowing: Approaches to Traffic Flow and Traffic Jam Modeling , 2003, Oper. Res..

[14]  B. Piccoli,et al.  Traffic Flow on a Road Network Using the Aw–Rascle Model , 2006 .

[15]  Axel Klar,et al.  A Kinetic Model for Vehicular Traffic Derived from a Stochastic Microscopic Model , 1996 .

[16]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[17]  CIRO D’APICE,et al.  Packet Flow on Telecommunication Networks , 2006, SIAM J. Math. Anal..

[18]  Denis Serre,et al.  Handbook of mathematical fluid dynamics , 2002 .

[19]  Nicola Bellomo,et al.  ON THE MATHEMATICAL THEORY OF VEHICULAR TRAFFIC FLOW I: FLUID DYNAMIC AND KINETIC MODELLING , 2002 .

[20]  P. Lax.  Hyperbolic systems of conservation laws , 2006 .

[21]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[22]  N. Bellomo,et al.  First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow , 2005 .

[23]  Robert Herman,et al.  Kinetic theory of vehicular traffic , 1971 .

[24]  Michel Rascle,et al.  Resurrection of "Second Order" Models of Traffic Flow , 2000, SIAM J. Appl. Math..

[25]  D. Gazis,et al.  Nonlinear Follow-the-Leader Models of Traffic Flow , 1961 .

[26]  Axel Klar,et al.  An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations , 2002 .

[27]  D. Serre.  Hyperbolicité, entropies, ondes de choc , 1996 .

[28]  Dirk Helbing.  From microscopic to macroscopic traffic models , 1998 .

[29]  Axel Klar,et al.  Derivation of Continuum Traffic Flow Models from Microscopic Follow-the-Leader Models , 2002, SIAM J. Appl. Math..

[30]  Wilhelm Leutzbach,et al.  Introduction to the Theory of Traffic Flow , 1987 .

[31]  Pierre Degond,et al.  A Model for the Formation and Evolution of Traffic Jams , 2008 .

[32]  Axel Klar,et al.  Multivalued Fundamental Diagrams and Stop and Go Waves for Continuum Traffic Flow Equations , 2004, SIAM J. Appl. Math..

[33]  S Bagnara,et al.  Trends in Occupational Health and Safety Policy in Italy , 1981, International journal of health services : planning, administration, evaluation.

[34]  Ida Bonzani,et al.  From experiments to hydrodynamic traffic flow models: I-modelling and parameter identification , 2003 .

[35]  Axel Klar,et al.  Enskog-like kinetic models for vehicular traffic , 1997 .

[36]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[37]  Axel Klar,et al.  Traffic flow: models and numerics , 2004 .

[38]  E. Ben-Naim,et al.  Kinetic Theory of Traffic Flows , 2003 .

[39]  Dirk Helbing,et al.  Memory effects in microscopic traffic models and wide scattering in flow-density data. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Nicola Bellomo,et al.  Generalized kinetic (Boltzmann) models: mathematical structures and applications , 2002 .

[41]  Ida Bonzani,et al.  Hydrodynamic models of traffic flow: Drivers' behaviour and nonlinear diffusion , 2000 .

[42]  James M. Greenberg,et al.  Extensions and Amplifications of a Traffic Model of Aw and Rascle , 2000, SIAM J. Appl. Math..

[43]  W. R. Buckland.  Theory of Traffic Flow , 1961 .

[44]  Ida Bonzani,et al.  Stochastic modelling of traffic flow , 2002 .

[45]  Partha Chakroborty,et al.  Microscopic Modeling of Driver Behavior in Uninterrupted Traffic Flow , 2004 .

[46]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[47]  H. Holden,et al.  A mathematical model of traffic flow on a network of unidirectional roads , 1995 .

[48]  Dirk Helbing,et al.  Delays, inaccuracies and anticipation in microscopic traffic models , 2006 .

[49]  Ciro D'Apice,et al.  A fluid dynamic model for supply chains , 2006, Networks Heterog. Media.

[50]  B. Kerner.  THE PHYSICS OF TRAFFIC , 1999 .

[51]  B. Kerner,et al.  A microscopic model for phase transitions in traffic flow , 2002 .

[52]  D. Serre,et al.  Hyperbolicity, entropies, shock waves , 1999 .

[53]  Dirk Helbing,et al.  Granular and Traffic Flow ’99: Social, Traffic, and Granular Dynamics , 2000 .

[54]  Axel Klar,et al.  Modelling and optimization of supply chains on complex networks , 2006 .

[55]  Serge P. Hoogendoorn,et al.  State-of-the-art of vehicular traffic flow modelling , 2001 .

[56]  Axel Klar,et al.  Gas flow in pipeline networks , 2006, Networks Heterog. Media.

[57]  Helbing,et al.  Congested traffic states in empirical observations and microscopic simulations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[58]  Mauro Garavello,et al.  Traffic Flow on Networks , 2006 .

[59]  Nicola Bellomo,et al.  Mathematical Topics In Nonlinear Kinetic Theory , 1988 .

[60]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[61]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[62]  R. Colombo.  A 2 × 2 hyperbolic traffic flow model , 2002 .

[63]  M. Rascle.  An improved macroscopic model of traffic flow: Derivation and links with the Lighthill-Whitham model , 2002 .

[64]  Andrea Tosin,et al.  Discrete kinetic and stochastic game theory for vehicular traffic: Modeling and mathematical problems , 2008 .

[65]  Eli Ben-Naim,et al.  Steady-state properties of traffic flows , 1998 .

[66]  Carlo Cercignani,et al.  On the kinetic theory of a dense gas of rough spheres , 1988 .

[67]  Mauro Garavello,et al.  Traffic Flow on a Road Network , 2005, SIAM J. Math. Anal..

[68]  Boris S. Kerner,et al.  Phase Transitions in Traffic Flow , 2000 .

[69]  Michael Herty,et al.  Optimization criteria for modelling intersections of vehicular traffic flow , 2006, Networks Heterog. Media.

[70]  Rinaldo M. Colombo,et al.  Hyperbolic Phase Transitions in Traffic Flow , 2003, SIAM J. Appl. Math..

[71]  Axel Klar,et al.  Kinetic Derivation of Macroscopic Anticipation Models for Vehicular Traffic , 2000, SIAM J. Appl. Math..

[72]  Yi Pan,et al.  A Study of Average-Case Speedup and Scalability of Parallel Computations on Static Networks , 1997, PDPTA.

[73]  S. L. Paveri-Fontana,et al.  On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis , 1975 .

[74]  Maria Letizia Bertotti,et al.  FROM DISCRETE KINETIC AND STOCHASTIC GAME THEORY TO MODELLING COMPLEX SYSTEMS IN APPLIED SCIENCES , 2004 .