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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 |
References:
[1] |
T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics, J. Stat. Phys, 133 (2008), 1083-1105. |
[2] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
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), 185-220.
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), 107-127.
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), 787-830.
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), 683-688.
doi: 10.1016/S0895-7177(02)80029-2. |
[7] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2003), 708-721.
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), 269-287.
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), 79-93.
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), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[11] |
C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B, 40 (2006), 396-403.
doi: 10.1016/j.trb.2005.05.004. |
[12] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 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, Berkeley, CA, 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), 056113, 13 pp.
doi: 10.1103/PhysRevE.79.056113. |
[16] |
H. Greenberg, An analysis of traffic flow, Oper. Res., 7 (1959), 79-85.
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), 729-745.
doi: 10.1137/S0036139900378657. |
[18] |
J. M. Greenberg, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185(electronic).
doi: 10.1137/S0036139903431737. |
[19] |
B. D. Greenshields, A study of traffic capacity, Proceedings of the Highway Research Record, 14 (1935), 448-477. |
[20] |
D. Helbing, Video of traffic waves, Website. http://www.trafficforum.org/stopandgo. |
[21] |
D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[22] |
R. Herman and I. Prigogine, "Kinetic Theory of Vehicular Traffic," Elsevier, New York, 1971. |
[23] |
R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12. |
[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), 3797-3800.
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), 4200-4216.
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), R2335-R2338.
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), 54-83.
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), 1749-1766.
doi: 10.1137/S0036139999356181. |
[30] |
T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), 5574-5582.
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), 218-221.
doi: 10.1103/PhysRevE.52.218. |
[32] |
J.-P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45. |
[33] |
R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 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), 1042-1061(electronic).
doi: 10.1137/S0036139999356788. |
[35] |
T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Comm. Math. Sci., 3 (2005), 101-118. |
[36] |
T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds, Discrete Contin. Dyn. Syst., 24 (2009), 511-521.
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), 317-345.
doi: 10.1098/rspa.1955.0089. |
[38] |
T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175.
doi: 10.1007/BF01210707. |
[39] |
A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks, Traffic Engrg. Control, 31 (1990), 466-470. |
[40] |
K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221-2229.
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, editor, Proceedings of the 14th International Symposium on Transportation and Trafic Theory, pages 51-79, Jerusalem, 1999. |
[42] |
G. F. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229.
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), 289-303.
doi: 10.1016/0191-2615(93)90039-D. |
[44] |
Minnesota Department of Transportation, Mn/DOT traffic data, Website. http://data.dot.state.mn.us/datatools. |
[45] |
H. J. Payne, Models of freeway traffic and control, Proc. Simulation Council, 1 (1971), 51-61. |
[46] |
H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffi, Transp. Res. Rec., 722 (1979), 68-77. |
[47] |
W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transportation Planning and Technology, 5 (1979), 131-138.
doi: 10.1080/03081067908717157. |
[48] |
L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281.
doi: 10.1063/1.1721265. |
[49] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
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), 1150-1162(electronic).
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), 033001.
doi: 10.1088/1367-2630/10/3/033001. |
[53] |
R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow, Technical report, Yale Bureau of Highway Traffic, 1961. |
[54] |
Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM), Website. http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. |
[55] |
P. Varaiya, Reducing highway congestion: An empirical approach, Eur. J. Control, 11 (2005), 301-309.
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), 141-167.
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), 158-171.
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), 113-158.
doi: 10.1002/cpa.3160120107. |
[59] |
G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. xvi+636 pp. |
[60] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
show all references
References:
[1] |
T. Alperovich and A. Sopasakis, Modeling highway traffic with stochastic dynamics, J. Stat. Phys, 133 (2008), 1083-1105. |
[2] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
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), 185-220.
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), 107-127.
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), 787-830.
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), 683-688.
doi: 10.1016/S0895-7177(02)80029-2. |
[7] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2003), 708-721.
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), 269-287.
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), 79-93.
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), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[11] |
C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B, 40 (2006), 396-403.
doi: 10.1016/j.trb.2005.05.004. |
[12] |
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 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, Berkeley, CA, 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), 056113, 13 pp.
doi: 10.1103/PhysRevE.79.056113. |
[16] |
H. Greenberg, An analysis of traffic flow, Oper. Res., 7 (1959), 79-85.
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), 729-745.
doi: 10.1137/S0036139900378657. |
[18] |
J. M. Greenberg, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185(electronic).
doi: 10.1137/S0036139903431737. |
[19] |
B. D. Greenshields, A study of traffic capacity, Proceedings of the Highway Research Record, 14 (1935), 448-477. |
[20] |
D. Helbing, Video of traffic waves, Website. http://www.trafficforum.org/stopandgo. |
[21] |
D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141.
doi: 10.1103/RevModPhys.73.1067. |
[22] |
R. Herman and I. Prigogine, "Kinetic Theory of Vehicular Traffic," Elsevier, New York, 1971. |
[23] |
R. Illner, A. Klar and T. Materne, Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12. |
[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), 3797-3800.
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), 4200-4216.
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), R2335-R2338.
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), 54-83.
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), 1749-1766.
doi: 10.1137/S0036139999356181. |
[30] |
T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), 5574-5582.
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), 218-221.
doi: 10.1103/PhysRevE.52.218. |
[32] |
J.-P. Lebacque, Les modeles macroscopiques du traffic, Annales des Ponts., 67 (1993), 24-45. |
[33] |
R. J. LeVeque, "Numerical Methods for Conservation Laws," Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 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), 1042-1061(electronic).
doi: 10.1137/S0036139999356788. |
[35] |
T. Li and H. Liu, Stability of a traffic flow model with nonconvex relaxation, Comm. Math. Sci., 3 (2005), 101-118. |
[36] |
T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds, Discrete Contin. Dyn. Syst., 24 (2009), 511-521.
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), 317-345.
doi: 10.1098/rspa.1955.0089. |
[38] |
T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175.
doi: 10.1007/BF01210707. |
[39] |
A. Messmer and M. Papageorgiou, METANET: A macroscopic simulation program for motorway networks, Traffic Engrg. Control, 31 (1990), 466-470. |
[40] |
K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Phys. I France, 2 (1992), 2221-2229.
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, editor, Proceedings of the 14th International Symposium on Transportation and Trafic Theory, pages 51-79, Jerusalem, 1999. |
[42] |
G. F. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229.
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), 289-303.
doi: 10.1016/0191-2615(93)90039-D. |
[44] |
Minnesota Department of Transportation, Mn/DOT traffic data, Website. http://data.dot.state.mn.us/datatools. |
[45] |
H. J. Payne, Models of freeway traffic and control, Proc. Simulation Council, 1 (1971), 51-61. |
[46] |
H. J. Payne, FREEFLO: A macroscopic simulation model of freeway traffi, Transp. Res. Rec., 722 (1979), 68-77. |
[47] |
W. F. Phillips, A kinetic model for traffic flow with continuum implications, Transportation Planning and Technology, 5 (1979), 131-138.
doi: 10.1080/03081067908717157. |
[48] |
L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281.
doi: 10.1063/1.1721265. |
[49] |
P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.
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), 1150-1162(electronic).
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), 033001.
doi: 10.1088/1367-2630/10/3/033001. |
[53] |
R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow, Technical report, Yale Bureau of Highway Traffic, 1961. |
[54] |
Federal Highway Administration US Department of Transportation, Next generation simulation (NGSIM), Website. http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. |
[55] |
P. Varaiya, Reducing highway congestion: An empirical approach, Eur. J. Control, 11 (2005), 301-309.
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), 141-167.
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), 158-171.
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), 113-158.
doi: 10.1002/cpa.3160120107. |
[59] |
G. B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. xvi+636 pp. |
[60] |
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
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