# American Institute of Mathematical Sciences

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

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

 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

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.
Citation: Florent Berthelin, Damien Broizat. A model for the evolution of traffic jams in multi-lane. Kinetic & 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. Google Scholar [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. Google Scholar [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. doi: 10.1103/PhysRevE.51.1035. Google Scholar [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. doi: 10.1137/090746677. Google Scholar [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. doi: 10.1142/S0218202502001635. Google Scholar [6] F. Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint,, Math. Mod. and Numer. Anal., 37 (2003), 479. Google Scholar [7] F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss,, Annales de l'Institut H. Poincaré, 20 (2003), 975. doi: 10.1016/S0294-1449(03)00012-X. Google Scholar [8] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, ARMA, 187 (2008), 185. doi: 10.1007/s00205-007-0061-9. Google Scholar [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. Google Scholar [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. Google Scholar [11] Y. Brenier and E. Grenier, Sticky particles and scalar conservations laws,, SIAM J. Numer. Anal., 35 (1998), 2317. Google Scholar [12] R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Math. Comp. Modeling, 35 (2002), 683. doi: 10.1016/S0895-7177(02)80029-2. Google Scholar [13] C. Daganzo, Requiem for second order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277. doi: 10.1016/0191-2615(95)00007-Z. Google Scholar [14] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation,, Kinetic and Related Models, 1 (2008), 279. Google Scholar [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. Google Scholar [16] D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545. Google Scholar [17] D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164. Google Scholar [18] D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. Google Scholar [19] A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic,, Surveys Math. Ind., 6 (1996), 215. Google Scholar [20] A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic,, J. Stat. Phys., (1997), 91. Google Scholar [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. Google Scholar [22] P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions,, Transp. Theory Stat. Phys., 24 (1995), 383. doi: 10.1080/00411459508205136. Google Scholar [23] H. J. Payne, "Models of Freeway Traffic and Control,", Simulation Council, (1971). Google Scholar [24] H. J. Payne, FREFLO: A macroscopic simulation model of freeway traffic,, Transportation Research Record, 722 (1979), 68. Google Scholar [25] I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow,, Theory of Traffic Flow, (1961), 158. Google Scholar [26] I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier publishing co, (1971). Google Scholar [27] P. I. Richards, Shock waves on a highway,, Operations Research, 4 (1956), 42. Google Scholar [28] M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Res. B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar

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. Google Scholar [2] A. Aw and M. Rascle, Resurrection of second order models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. Google Scholar [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. doi: 10.1103/PhysRevE.51.1035. Google Scholar [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. doi: 10.1137/090746677. Google Scholar [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. doi: 10.1142/S0218202502001635. Google Scholar [6] F. Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint,, Math. Mod. and Numer. Anal., 37 (2003), 479. Google Scholar [7] F. Berthelin and F. Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss,, Annales de l'Institut H. Poincaré, 20 (2003), 975. doi: 10.1016/S0294-1449(03)00012-X. Google Scholar [8] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, ARMA, 187 (2008), 185. doi: 10.1007/s00205-007-0061-9. Google Scholar [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. Google Scholar [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. Google Scholar [11] Y. Brenier and E. Grenier, Sticky particles and scalar conservations laws,, SIAM J. Numer. Anal., 35 (1998), 2317. Google Scholar [12] R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Math. Comp. Modeling, 35 (2002), 683. doi: 10.1016/S0895-7177(02)80029-2. Google Scholar [13] C. Daganzo, Requiem for second order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277. doi: 10.1016/0191-2615(95)00007-Z. Google Scholar [14] P. Degond and M. Delitala, Modelling and simulation of vehicular traffic jam formation,, Kinetic and Related Models, 1 (2008), 279. Google Scholar [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. Google Scholar [16] D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545. Google Scholar [17] D. Helbing, Improved fluid-dynamic model for vehicular traffic,, Phys. Rev. E, 51 (1995), 3164. doi: 10.1103/PhysRevE.51.3164. Google Scholar [18] D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. Google Scholar [19] A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic,, Surveys Math. Ind., 6 (1996), 215. Google Scholar [20] A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic,, J. Stat. Phys., (1997), 91. Google Scholar [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. Google Scholar [22] P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions,, Transp. Theory Stat. Phys., 24 (1995), 383. doi: 10.1080/00411459508205136. Google Scholar [23] H. J. Payne, "Models of Freeway Traffic and Control,", Simulation Council, (1971). Google Scholar [24] H. J. Payne, FREFLO: A macroscopic simulation model of freeway traffic,, Transportation Research Record, 722 (1979), 68. Google Scholar [25] I. Prigogine, A Boltzmann-like approach to the statistical theory of traffic flow,, Theory of Traffic Flow, (1961), 158. Google Scholar [26] I. Prigogine and R. Herman, "Kinetic Theory of Vehicular Traffic,", American Elsevier publishing co, (1971). Google Scholar [27] P. I. Richards, Shock waves on a highway,, Operations Research, 4 (1956), 42. Google Scholar [28] M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transportation Res. B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3. Google Scholar
 [1] Cécile Appert-Rolland, Pierre Degond, Sébastien Motsch. Two-way multi-lane traffic model for pedestrians in corridors. Networks & Heterogeneous Media, 2011, 6 (3) : 351-381. doi: 10.3934/nhm.2011.6.351 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431 [9] Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 599-630. doi: 10.3934/dcdsb.2005.5.599 [17] Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393 [18] Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681 [19] Florent Berthelin, Paola Goatin. Regularity results for the solutions of a non-local model of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3197-3213. doi: 10.3934/dcds.2019132 [20] Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135

2018 Impact Factor: 1.38