# American Institute of Mathematical Sciences

September  2013, 8(3): 745-772. doi: 10.3934/nhm.2013.8.745

## Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models

 1 Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122 2 Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada 3 4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900, Saudi Arabia 4 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Received  April 2012 Revised  June 2013 Published  October 2013

Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
Citation: Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745
##### References:
 [1] T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099. [3] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Ration. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9. [4] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107. doi: 10.1137/090754467. [5] G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy,, Comm. Pure Appl. Math., 47 (1994), 787. doi: 10.1002/cpa.3160470602. [6] R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Traffic flow—modelling and simulation. Math. Comput. Modelling, 35 (2002), 683. doi: 10.1016/S0895-7177(02)80029-2. [7] R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708. doi: 10.1137/S0036139901393184. [8] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transp. Res. B, 28 (1994), 269. doi: 10.1016/0191-2615(94)90002-7. [9] C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transp. Res. B, 29 (1995), 79. doi: 10.1016/0191-2615(94)00022-R. [10] C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277. doi: 10.1016/0191-2615(95)00007-Z. [11] C. F. Daganzo, In traffic flow, cellular automata = kinematic waves,, Transp. Res. B, 40 (2006), 396. doi: 10.1016/j.trb.2005.05.004. [12] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998). [13] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model,, in preparation, (2012). [14] W. Fickett and W. C. Davis, "Detonation,", Univ. of California Press, (1979). [15] M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.056113. [16] H. Greenberg, An analysis of traffic flow,, Oper. Res., 7 (1959), 79. doi: 10.1287/opre.7.1.79. [17] J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657. [18] J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737. [19] B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448. [20] D. Helbing, Video of traffic waves,, Website. , (). [21] D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. [22] R. Herman and I. Prigogine, "Kinetic Theory of Vehicular Traffic,", Elsevier, (1971). [23] R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1. [24] A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2013). [25] B. S. Kerner, Experimental features of self-organization in traffic flow,, Phys. Rev. Lett., 81 (1998), 3797. doi: 10.1103/PhysRevLett.81.3797. [26] B. S. Kerner, S. L. Klenov and P. Konhäuser, Asymptotic theory of traffic jams,, Phys. Rev. E, 56 (1997), 4200. doi: 10.1103/PhysRevE.56.4200. [27] B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993). doi: 10.1103/PhysRevE.48.R2335. [28] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54. [29] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181. [30] T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion,, Phys. Rev. E, 52 (1995), 5574. doi: 10.1103/PhysRevE.52.5574. [31] D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218. doi: 10.1103/PhysRevE.52.218. [32] J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24. [33] R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992). doi: 10.1007/978-3-0348-8629-1. [34] T. Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042. doi: 10.1137/S0036139999356788. [35] T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101. [36] T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete Contin. Dyn. Syst., 24 (2009), 511. doi: 10.3934/dcds.2009.24.511. [37] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089. [38] T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707. [39] A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks,, Traffic Engrg. Control, 31 (1990), 466. [40] K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221. doi: 10.1051/jp1:1992277. [41] P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, In A. Ceder, (1999), 51. [42] G. F. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209. doi: 10.1287/opre.9.2.209. [43] G. F. Newell, A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289. doi: 10.1016/0191-2615(93)90039-D. [44] Minnesota Department of Transportation, Mn/DOT traffic data,, Website. , (). [45] H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51. [46] H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffi,, Transp. Res. Rec., 722 (1979), 68. [47] W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131. doi: 10.1080/03081067908717157. [48] L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274. doi: 10.1063/1.1721265. [49] P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. [50] B. Seibold, R. R. Rosales, M. R. Flynn and A. R. Kasimov, Classification of traveling wave solutions of the inhomogeneous Aw-Rascle-Zhang model,, in preparation, (2013). [51] F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150. doi: 10.1137/050627113. [52] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks - Experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/3/033001. [53] R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow,, Technical report, (1961). [54] Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM),, Website. , (). [55] P. Varaiya, Reducing highway congestion: An empirical approach,, Eur. J. Control, 11 (2005), 301. doi: 10.3166/ejc.11.301-309. [56] Y. Wang and M. Papageorgiou, Real-time freeway traffic state estimation based on extended Kalman filter: A general approach,, Transp. Res. B, 39 (2005), 141. doi: 10.1016/j.trb.2004.03.003. [57] J. G. Wardrop and G. Charlesworth, A method of estimating speed and flow of traffic from a moving vehicle,, Proc. Instn. Civ. Engrs., 3 (1954), 158. doi: 10.1680/ipeds.1954.11628. [58] G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics,, Comm. Pure Appl. Math., 12 (1959), 113. doi: 10.1002/cpa.3160120107. [59] G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974). [60] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transp. Res. B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3.

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##### References:
 [1] T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics,, J. Stat. Phys, 133 (2008), 1083. [2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow,, SIAM J. Appl. Math., 60 (2000), 916. doi: 10.1137/S0036139997332099. [3] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams,, Arch. Ration. Mech. Anal., 187 (2008), 185. doi: 10.1007/s00205-007-0061-9. [4] S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen, A general phase transition model for vehicular traffic,, SIAM J. Appl. Math., 71 (2011), 107. doi: 10.1137/090754467. [5] G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy,, Comm. Pure Appl. Math., 47 (1994), 787. doi: 10.1002/cpa.3160470602. [6] R. M. Colombo, On a $2\times 2$ hyperbolic traffic flow model,, Traffic flow—modelling and simulation. Math. Comput. Modelling, 35 (2002), 683. doi: 10.1016/S0895-7177(02)80029-2. [7] R. M. Colombo, Hyperbolic phase transitions in traffic flow,, SIAM J. Appl. Math., 63 (2003), 708. doi: 10.1137/S0036139901393184. [8] C. F. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory,, Transp. Res. B, 28 (1994), 269. doi: 10.1016/0191-2615(94)90002-7. [9] C. F. Daganzo, The cell transmission model, part II: Network traffic,, Transp. Res. B, 29 (1995), 79. doi: 10.1016/0191-2615(94)00022-R. [10] C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow,, Transp. Res. B, 29 (1995), 277. doi: 10.1016/0191-2615(95)00007-Z. [11] C. F. Daganzo, In traffic flow, cellular automata = kinematic waves,, Transp. Res. B, 40 (2006), 396. doi: 10.1016/j.trb.2005.05.004. [12] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998). [13] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model,, in preparation, (2012). [14] W. Fickett and W. C. Davis, "Detonation,", Univ. of California Press, (1979). [15] M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales and B. Seibold, Self-sustained nonlinear waves in traffic flow,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.056113. [16] H. Greenberg, An analysis of traffic flow,, Oper. Res., 7 (1959), 79. doi: 10.1287/opre.7.1.79. [17] J. M. Greenberg, Extension and amplification of the Aw-Rascle model,, SIAM J. Appl. Math., 62 (2001), 729. doi: 10.1137/S0036139900378657. [18] J. M. Greenberg, Congestion redux,, SIAM J. Appl. Math., 64 (2004), 1175. doi: 10.1137/S0036139903431737. [19] B. D. Greenshields, A study of traffic capacity,, Proceedings of the Highway Research Record, 14 (1935), 448. [20] D. Helbing, Video of traffic waves,, Website. , (). [21] D. Helbing, Traffic and related self-driven many-particle systems,, Reviews of Modern Physics, 73 (2001), 1067. doi: 10.1103/RevModPhys.73.1067. [22] R. Herman and I. Prigogine, "Kinetic Theory of Vehicular Traffic,", Elsevier, (1971). [23] R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow,, Commun. Math. Sci., 1 (2003), 1. [24] A. R. Kasimov, R. R. Rosales, B. Seibold and M. R. Flynn, Existence of jamitons in hyperbolic relaxation systems with application to traffic flow,, in preparation, (2013). [25] B. S. Kerner, Experimental features of self-organization in traffic flow,, Phys. Rev. Lett., 81 (1998), 3797. doi: 10.1103/PhysRevLett.81.3797. [26] B. S. Kerner, S. L. Klenov and P. Konhäuser, Asymptotic theory of traffic jams,, Phys. Rev. E, 56 (1997), 4200. doi: 10.1103/PhysRevE.56.4200. [27] B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow,, Phys. Rev. E, 48 (1993). doi: 10.1103/PhysRevE.48.R2335. [28] B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow,, Phys. Rev. E, 50 (1994), 54. doi: 10.1103/PhysRevE.50.54. [29] A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic,, SIAM J. Appl. Math., 60 (2000), 1749. doi: 10.1137/S0036139999356181. [30] T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion,, Phys. Rev. E, 52 (1995), 5574. doi: 10.1103/PhysRevE.52.5574. [31] D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection,, Phys. Rev. E, 52 (1995), 218. doi: 10.1103/PhysRevE.52.218. [32] J.-P. Lebacque, Les modeles macroscopiques du traffic,, Annales des Ponts., 67 (1993), 24. [33] R. J. LeVeque, "Numerical Methods for Conservation Laws,", Second edition, (1992). doi: 10.1007/978-3-0348-8629-1. [34] T. Li, Global solutions and zero relaxation limit for a traffic flow model,, SIAM J. Appl. Math., 61 (2000), 1042. doi: 10.1137/S0036139999356788. [35] T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation,, Comm. Math. Sci., 3 (2005), 101. [36] T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds,, Discrete Contin. Dyn. Syst., 24 (2009), 511. doi: 10.3934/dcds.2009.24.511. [37] M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads,, Proc. Roy. Soc. A, 229 (1955), 317. doi: 10.1098/rspa.1955.0089. [38] T. P. Liu, Hyperbolic conservation laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707. [39] A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks,, Traffic Engrg. Control, 31 (1990), 466. [40] K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic,, J. Phys. I France, 2 (1992), 2221. doi: 10.1051/jp1:1992277. [41] P. Nelson and A. Sopasakis, The Chapman-Enskog expansion: A novel approach to hierarchical extension of Lighthill-Whitham models,, In A. Ceder, (1999), 51. [42] G. F. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209. doi: 10.1287/opre.9.2.209. [43] G. F. Newell, A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks,, Transp. Res. B, 27 (1993), 289. doi: 10.1016/0191-2615(93)90039-D. [44] Minnesota Department of Transportation, Mn/DOT traffic data,, Website. , (). [45] H. J. Payne, Models of freeway traffic and control,, Proc. Simulation Council, 1 (1971), 51. [46] H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffi,, Transp. Res. Rec., 722 (1979), 68. [47] W. F. Phillips, A kinetic model for traffic flow with continuum implications,, Transportation Planning and Technology, 5 (1979), 131. doi: 10.1080/03081067908717157. [48] L. A. Pipes, An operational analysis of traffic dynamics,, Journal of Applied Physics, 24 (1953), 274. doi: 10.1063/1.1721265. [49] P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42. doi: 10.1287/opre.4.1.42. [50] B. Seibold, R. R. Rosales, M. R. Flynn and A. R. Kasimov, Classification of traveling wave solutions of the inhomogeneous Aw-Rascle-Zhang model,, in preparation, (2013). [51] F. Siebel and W. Mauser, On the fundamental diagram of traffic flow,, SIAM J. Appl. Math., 66 (2006), 1150. doi: 10.1137/050627113. [52] Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S. Tadaki and S. Yukawa, Traffic jams without bottlenecks - Experimental evidence for the physical mechanism of the formation of a jam,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/3/033001. [53] R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow,, Technical report, (1961). [54] Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM),, Website. , (). [55] P. Varaiya, Reducing highway congestion: An empirical approach,, Eur. J. Control, 11 (2005), 301. doi: 10.3166/ejc.11.301-309. [56] Y. Wang and M. Papageorgiou, Real-time freeway traffic state estimation based on extended Kalman filter: A general approach,, Transp. Res. B, 39 (2005), 141. doi: 10.1016/j.trb.2004.03.003. [57] J. G. Wardrop and G. Charlesworth, A method of estimating speed and flow of traffic from a moving vehicle,, Proc. Instn. Civ. Engrs., 3 (1954), 158. doi: 10.1680/ipeds.1954.11628. [58] G. B. Whitham, Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics,, Comm. Pure Appl. Math., 12 (1959), 113. doi: 10.1002/cpa.3160120107. [59] G. B. Whitham, "Linear and Nonlinear Waves,", Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], (1974). [60] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,, Transp. Res. B, 36 (2002), 275. doi: 10.1016/S0191-2615(00)00050-3.
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