Networks and Heterogeneous Media (NHM)

Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models
Pages: 745 - 772, Issue 3, September 2013

doi:10.3934/nhm.2013.8.745      Abstract        References        Full text (1439.2K)                  Related Articles

Benjamin Seibold - Temple University, Department of Mathematics, 1805 North Broad Street Philadelphia, PA 19122, United States (email)
Morris R. Flynn - Department of Mechanical Engineering, University of Alberta, Edmonton, AB, T6G 2G8, Canada (email)
Aslan R. Kasimov - 4700 King Abdullah University of, Science and Technology, Thuwal 23955-6900, Saudi Arabia (email)
Rodolfo R. Rosales - Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, United States (email)

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.       
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.       
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.       
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.       
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.       
7 R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2003), 708-721.       
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.
9 C. F. Daganzo, The cell transmission model, part II: Network traffic, Transp. Res. B, 29 (1995), 79-93.
10 C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transp. Res. B, 29 (1995), 277-286.
11 C. F. Daganzo, In traffic flow, cellular automata = kinematic waves, Transp. Res. B, 40 (2006), 396-403.
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.       
16 H. Greenberg, An analysis of traffic flow, Oper. Res., 7 (1959), 79-85.       
17 J. M. Greenberg, Extension and amplification of the Aw-Rascle model, SIAM J. Appl. Math., 62 (2001), 729-745.       
18 J. M. Greenberg, Congestion redux, SIAM J. Appl. Math., 64 (2004), 1175-1185(electronic).       
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.
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.
26 B. S. Kerner, S. L. Klenov and P. Konhäuser, Asymptotic theory of traffic jams, Phys. Rev. E, 56 (1997), 4200-4216.
27 B. S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E, 48 (1993), R2335-R2338.
28 B. S. Kerner and P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54-83.
29 A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766.       
30 T. S. Komatsu and S. Sasa, Kink soliton characterizing traffic congestion, Phys. Rev. E, 52 (1995), 5574-5582.
31 D. A. Kurtze and D. C. Hong, Traffic jams, granular flow, and soliton selection, Phys. Rev. E, 52 (1995), 218-221.
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.       
34 T. Li, Global solutions and zero relaxation limit for a traffic flow model, SIAM J. Appl. Math., 61 (2000), 1042-1061(electronic).       
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.       
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.       
38 T. P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175.       
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.
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.       
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.
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.
48 L. A. Pipes, An operational analysis of traffic dynamics, Journal of Applied Physics, 24 (1953), 274-281.       
49 P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51.       
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).       
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.
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.       
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.
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.
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.       
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.

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