A 2-Phase Traffic Model Based on a Speed Bound

We extend the classical Lighthill–Whitham–Richards (LWR) traffic model allowing different maximal speeds for different vehicles. Then we add a uniform bound on the traffic speed. The result, presented in this paper, is a new macroscopic model displaying two phases based on a nonsmooth $2\times2$ system of conservation laws. This model is compared with other models of the same type in the current literature, as well as with a kinetic one. Moreover, we establish a rigorous connection between a microscopic follow-the-leader model based on ordinary differential equations and this macroscopic continuum model.

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

[2]  Georges Bastin,et al.  A second order model of road junctions in fluid models of traffic networks , 2007, Networks Heterog. Media.

[3]  B. Temple Systems of conservation laws with invariant submanifolds , 1983 .

[4]  R. Colombo,et al.  Measure valued solutions to conservation laws motivated by traffic modelling , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[6]  J. Lebacque,et al.  Generic Second Order Traffic Flow Modelling , 2007 .

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

[8]  A. Bressan Hyperbolic Systems of Conservation Laws , 1999 .

[9]  Rinaldo M. Colombo,et al.  On the role of source terms in continuum traffic flow models , 2006, Math. Comput. Model..

[10]  Jean Patrick Lebacque,et al.  Modélisation du trafic autoroutier au second ordre , 2008 .

[11]  Wen Shen,et al.  Global Existence of Large BV Solutions in a Model of Granular Flow , 2009 .

[12]  Kara M. Kockelman Modeling traffic's flow-density relation: Accommodation of multiple flow regimes and traveler types , 2001 .

[13]  Wolfram Mauser,et al.  On the Fundamental Diagram of Traffic Flow , 2006, SIAM J. Appl. Math..

[14]  Patrizia Bagnerini,et al.  A Multiclass Homogenized Hyperbolic Model of Traffic Flow , 2003, SIAM J. Math. Anal..

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

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

[17]  K. Hadeler,et al.  Dynamical models for granular matter , 1999 .

[18]  M. Omizo,et al.  Modeling , 1983, Encyclopedic Dictionary of Archaeology.

[19]  Harald Hanche-Olsen,et al.  Car-following and the macroscopic Aw-Rascle traffic flow model , 2009 .

[20]  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.

[21]  Michael J. Cassidy,et al.  Relation between traffic density and capacity drop at three freeway bottlenecks , 2007 .

[22]  Paola Goatin,et al.  The Aw-Rascle vehicular traffic flow model with phase transitions , 2006, Math. Comput. Model..

[23]  Salissou Moutari,et al.  A Hybrid Lagrangian Model Based on the Aw--Rascle Traffic Flow Model , 2007, SIAM J. Appl. Math..

[24]  Rinaldo M. Colombo,et al.  An $n$-populations model for traffic flow , 2003, European Journal of Applied Mathematics.

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

[26]  Agustinus Peter Sahanggamu Phase transitions of traffic flow , 2010 .

[27]  L. C. Edie Car-Following and Steady-State Theory for Noncongested Traffic , 1961 .

[28]  Barbara Lee Keyfitz,et al.  A system of non-strictly hyperbolic conservation laws arising in elasticity theory , 1980 .

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

[30]  D. Hoff Invariant regions for systems of conservation laws , 1985 .

[31]  Fabio S. Priuli,et al.  Global well posedness of traffic flow models with phase transitions , 2007 .

[32]  Rui Jiang,et al.  Synchronized flow and phase separations in single-lane mixed traffic flow , 2007 .

[33]  Harald Hanche-Olsen,et al.  EXISTENCE OF SOLUTIONS FOR THE AW-RASCLE TRAFFIC FLOW MODEL WITH VACUUM , 2008 .

[34]  Dirk Helbing,et al.  Critical Discussion of "Synchronized Flow" , 2002 .

[35]  A. Klar,et al.  Congestion on Multilane Highways , 2002, SIAM J. Appl. Math..

[36]  Dong Ngoduy,et al.  Multiclass first-order simulation model to explain non-linear traffic phenomena , 2007 .

[37]  Axel Klar,et al.  Modeling, Simulation, and Optimization of Traffic Flow Networks , 2003, SIAM J. Sci. Comput..

[38]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[39]  Dirk Helbing,et al.  Understanding widely scattered traffic flows, the capacity drop, and platoons as effects of variance-driven time gaps. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Rinaldo M. Colombo,et al.  A Hölder continuous ODE related to traffic flow , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.