A Hybrid Lagrangian Model Based on the Aw--Rascle Traffic Flow Model

In this paper, we propose a simple fully discrete hybrid model for vehicular traffic flow, for which both the macroscopic and the microscopic models are based on a Lagrangian discretization of the Aw–Rascle (AR) model [A. Aw and M. Rascle, SIAM J. Appl. Math., 60 (2000), pp. 916–938]. This hybridization makes use of the relation between the AR macroscopic model and a follow-the-leader-type model [D. C. Gazis, R. Herman, and R. W. Rothery, Oper. Res., 9 (1961), pp. 545–567; R. Herman and I. Prigogine, Kinetic Theory of Vehicular Traffic, American Elsevier, New York, 1971], established in [A. Aw, A. Klar, M. Materne, and M. Rascle, SIAM J. Appl. Math., 63 (2002), pp. 259–278]. Moreover, in the hybrid model, the total variation in space of the velocity v is nonincreasing, the total variation in space of the specific volume $\tau$ is bounded, and the total variations in time of v and $\tau$ are bounded. Finally, we present some numerical simulations which confirm that the models' synchronization processes do ...

[1]  Carlos F. Daganzo,et al.  Lane-changing in traffic streams , 2006 .

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

[3]  Jean-François Gabard,et al.  Towards an hybrid macro-micro traffic flow simulation model , 2000 .

[4]  C. Daganzo.  THE CELL TRANSMISSION MODEL.. , 1994 .

[5]  Dirk Helbing,et al.  Macroscopic Simulation of Open Systems and Micro-Macro Link , 2000 .

[6]  H. M. Zhang.  A NON-EQUILIBRIUM TRAFFIC MODEL DEVOID OF GAS-LIKE BEHAVIOR , 2002 .

[7]  Brenna Argall,et al.  A Rigorous Treatment of a Follow-the-Leader Traffic Model with Traffic Lights Present , 2003, SIAM J. Appl. Math..

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

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

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

[11]  E. Bourrel,et al.  Mixing Micro and Macro Representations of Traffic Flow: a Hybrid Model Based on the LWR Theory , 2003 .

[12]  Dirk Helbing,et al.  Micro- and macro-simulation of freeway traffic , 2002 .

[13]  Jean-Baptiste Lesort,et al.  Mixing Microscopic and Macroscopic Representations of Traffic Flow: Hybrid Model Based on Lighthill–Whitham–Richards Theory , 2003 .

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

[15]  M. R. C. McDowell,et al.  Kinetic Theory of Vehicular Traffic , 1972 .

[16]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

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

[18]  Tong Li.  Well-posedness theory of an inhomogeneous traffic flow model , 2002 .

[19]  Michael Herty,et al.  Coupling Conditions for a Class of Second-Order Models for Traffic Flow , 2006, SIAM J. Math. Anal..

[20]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

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

[22]  Harold J Payne,et al.  FREFLO: A MACROSCOPIC SIMULATION MODEL OF FREEWAY TRAFFIC , 1979 .

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

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

[25]  P. Bagnerini,et al.  A Multi-Class Homogenized Hyperbolic Model of Tra c Flow , 2003 .

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

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

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

[29]  Gui-Qiang G. Chen,et al.  Propagation and cancellation of oscillations for hyperbolic systems of conservation laws , 1991 .

[30]  Haris N. Koutsopoulos,et al.  Hybrid Mesoscopic-Microscopic Traffic Simulation , 2004 .

[31]  J. P. Lebacque.  Les modèles macroscopiques de trafic , 1993 .

[32]  J. Lebacque.  THE GODUNOV SCHEME AND WHAT IT MEANS FOR FIRST ORDER TRAFFIC FLOW MODELS , 1996 .

[33]  Alberto Bressan,et al.  On the convergence of Godunov scheme for nonlinear hyperbolic systems , 2000 .

[34]  Shing Chung Josh Wong,et al.  A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers , 2002 .

[35]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

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

[37]  R. J. Diperna,et al.  Measure-valued solutions to conservation laws , 1985 .

[38]  R. Natalini.  Convergence to equilibrium for the relaxation approximations of conservation laws , 1996 .

[39]  Mathematisches Forschungsinstitut Oberwolfach,et al.  Hyperbolic Conservation Laws , 2004 .

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

[41]  David H. Wagner,et al.  Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions , 1987 .