A model for the evolution of traffic jams in multi-lane

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December 2012, 5(4): 697-728. doi: 10.3934/krm.2012.5.697

A model for the evolution of traffic jams in multi-lane

Florent Berthelin ^1, and Damien Broizat ^1,

1.	Laboratoire J. A. Dieudonné, UMR 7351 CNRS, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France, France

Received January 2012 Revised May 2012 Published November 2012

Abstract
Related Papers

In [8], Berthelin, Degond, Delitala and Rascle introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model consists of a Pressureless Gas Dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. In the present paper we propose an improvement of this model by allowing the road to be multi-lane piecewise. The idea is to use the maximal constraint to model the number of lanes. We also add in the model a parameter $\alpha$ which model the various speed limitations according to the number of lanes. We present the dynamical behaviour of clusters (traffic jams) and by approximation with such solutions, we obtain an existence result of weak solutions for any initial data.

Keywords: Traffic flow models, weak solutions, constrained pressureless gas dynamics, traffic jams., multi-lane.

Mathematics Subject Classification: Primary: 90B20, 35L60, 35L65; Secondary: 35L67, 35R99, 76L0.

Citation: Florent Berthelin, Damien Broizat. A model for the evolution of traffic jams in multi-lane. Kinetic and Related Models, 2012, 5 (4) : 697-728. doi: 10.3934/krm.2012.5.697

References:

[1]	A. Aw, A. Klar, A. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.
[2]	A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
[3]	M. Bando, K. Hesebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.
[4]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.
[5]	F. Berthelin, Existence and weak stability for a two-phase model with unilateral constraint, Math. Models & Methods in the Applied Sciences, 12 (2002), 249-272. doi: 10.1142/S0218202502001635.
[6]	F. Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint, Math. Mod. and Numer. Anal., 37 (2003), 479-494.
[7]	F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss, Annales de l'Institut H. Poincaré, Anal. non linéaire, 20 (2003), 975-997. doi: 10.1016/S0294-1449(03)00012-X.
[8]	F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, ARMA, Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9.
[9]	F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci. suppl., 18 (2008), 1269-1298.
[10]	F. Bouchut, Y. Brenier, J. Cortes and J. F. Ripoll, A hierarchy of models for two-phase flows, J. Nonlinear Science, 10 (2000), 639-660.
[11]	Y. Brenier and E. Grenier, Sticky particles and scalar conservations laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.
[12]	R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model, Math. Comp. Modeling, 35 (2002), 683-688. doi: 10.1016/S0895-7177(02)80029-2.
[13]	C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.
[14]	P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinetic and Related Models, 1 (2008), 279-293.
[15]	W. E, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380.
[16]	D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567.
[17]	D. Helbing, Improved fluid-dynamic model for vehicular traffic, Phys. Rev. E, 51 (1995), 3164-3169. doi: 10.1103/PhysRevE.51.3164.
[18]	D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
[19]	A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic, Surveys Math. Ind., 6 (1996), 215-239.
[20]	A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., (1997), 91-114.
[21]	M. J. Lighthill and J. B. Whitham, On kinematic waves. I: flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 1749-1766.
[22]	P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, Transp. Theory Stat. Phys., 24 (1995), 383-408. doi: 10.1080/00411459508205136.
[23]	H. J. Payne, "Models of Freeway Traffic and Control," Simulation Council, New York, 1971.
[24]	H. J. Payne, FREFLO: A macroscopic simulation model of freeway traffic, Transportation Research Record, 722 (1979), 68-75.
[25]	I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, Theory of Traffic Flow, (ed. R. Herman), (1961), 158-164.
[26]	I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic," American Elsevier publishing co, New York, 1971.
[27]	P. I. Richards, Shock waves on a highway, Operations Research, 4 (1956), 42-51.
[28]	M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. B, 36 (2002), 275-298. doi: 10.1016/S0191-2615(00)00050-3.

show all references

References:

[1]	A. Aw, A. Klar, A. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.
[2]	A. Aw and M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
[3]	M. Bando, K. Hesebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035.
[4]	N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463. doi: 10.1137/090746677.
[5]	F. Berthelin, Existence and weak stability for a two-phase model with unilateral constraint, Math. Models & Methods in the Applied Sciences, 12 (2002), 249-272. doi: 10.1142/S0218202502001635.
[6]	F. Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint, Math. Mod. and Numer. Anal., 37 (2003), 479-494.
[7]	F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss, Annales de l'Institut H. Poincaré, Anal. non linéaire, 20 (2003), 975-997. doi: 10.1016/S0294-1449(03)00012-X.
[8]	F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, ARMA, Arch. Ration. Mech. Anal., 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9.
[9]	F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, M. Rascle and J. Royer, A traffic-flow model with constraints for the modeling of traffic jams, Math. Models Methods Appl. Sci. suppl., 18 (2008), 1269-1298.
[10]	F. Bouchut, Y. Brenier, J. Cortes and J. F. Ripoll, A hierarchy of models for two-phase flows, J. Nonlinear Science, 10 (2000), 639-660.
[11]	Y. Brenier and E. Grenier, Sticky particles and scalar conservations laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328.
[12]	R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model, Math. Comp. Modeling, 35 (2002), 683-688. doi: 10.1016/S0895-7177(02)80029-2.
[13]	C. Daganzo, Requiem for second order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286. doi: 10.1016/0191-2615(95)00007-Z.
[14]	P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinetic and Related Models, 1 (2008), 279-293.
[15]	W. E, Y. G. Rykov and Y. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380.
[16]	D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567.
[17]	D. Helbing, Improved fluid-dynamic model for vehicular traffic, Phys. Rev. E, 51 (1995), 3164-3169. doi: 10.1103/PhysRevE.51.3164.
[18]	D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
[19]	A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic, Surveys Math. Ind., 6 (1996), 215-239.
[20]	A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., (1997), 91-114.
[21]	M. J. Lighthill and J. B. Whitham, On kinematic waves. I: flow movement in long rivers. II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 1749-1766.
[22]	P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, Transp. Theory Stat. Phys., 24 (1995), 383-408. doi: 10.1080/00411459508205136.
[23]	H. J. Payne, "Models of Freeway Traffic and Control," Simulation Council, New York, 1971.
[24]	H. J. Payne, FREFLO: A macroscopic simulation model of freeway traffic, Transportation Research Record, 722 (1979), 68-75.
[25]	I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow, Theory of Traffic Flow, (ed. R. Herman), (1961), 158-164.
[26]	I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic," American Elsevier publishing co, New York, 1971.
[27]	P. I. Richards, Shock waves on a highway, Operations Research, 4 (1956), 42-51.
[28]	M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. B, 36 (2002), 275-298. doi: 10.1016/S0191-2615(00)00050-3.

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