A general phase transition model for vehicular traffic

SIAM Journal on Applied Mathematics

Volumn 71, Issue 1, 2011, Pages 107-127

  Blandin, S ^a   Work, D ^b   Goatin, P ^c   Piccoli, B ^d   Bayen, A ^a  

^a Berkeley   (United States)

^b University of Illinois at Urbana Champaign   (United States)

^c INRIA   (France)

^d RUTGERS UNIVERSITY   (United States)

Author keywords

Hyperbolic systems of conservation laws; Macroscopic highway traffic flow model; Numerical scheme; Partial differential equations; Phase transition; Riemann solver

Indexed keywords

CAUCHY PROBLEMS; FLUX FUNCTION; FUNDAMENTAL DIAGRAM; GODUNOV SCHEME; HYPERBOLIC SYSTEMS OF CONSERVATION LAWS; MACROSCOPIC HIGHWAY TRAFFIC FLOW MODEL; NUMERICAL SCHEME; NUMERICAL SOLUTION; PHASE TRANSITION MODEL; RIEMANN PROBLEM; RIEMANN SOLVER; VEHICULAR TRAFFIC; WAVEFRONT TRACKING;

DIFFERENTIATION (CALCULUS); PARTIAL DIFFERENTIAL EQUATIONS; PHASE TRANSITIONS; PHYSICAL PROPERTIES; TRAFFIC CONTROL; TRAFFIC SURVEYS;

PHASE DIAGRAMS;

EID: 79952343024     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/090754467     Document Type: Article

References (50)

1. A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916-938.
2. C. Bardos, A.-Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017-1034.
3. S. Blandin, G. Bretti, A. Cutolo, and B. Piccoli, Numerical simulations of traffic data via fluid dynamic approach, Appl. Math. Comput., 210 (2009), pp. 441-454.
4. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford Lecture Ser. Math. Appl. 20, Oxford University Press, Oxford, UK, 2000.
5. M. Cassidy and J. Windover, Methodology for assessing dynamics of freeway traffic flow, Transportation Research Record, 1484 (1995), pp. 73-79.
6. C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), pp. 195-219.
7. R. Colombo, On a 2 × 2 hyperbolic traffic flow model, Math. Comput. Modelling, 35 (2002), pp. 683-688.
8. R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), pp. 708-721.
9. R. Colombo, P. Goatin, and F. Priuli, Global well-posedness of traffic flow models with phase transitions, Nonlinear Anal., 66 (2007), pp. 2413-2426. (Pubitemid 46349437)
10. C. Daganzo, The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory, Transportation Res. Part B, 28 (1994), pp. 269-287.
11. C. Daganzo, The cell transmission model, Part II: Network traffic, Transportation Res. Part B, 29 (1995), pp. 79-93.
12. C. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Res. Part B, 29 (1995), pp. 277-286. (Pubitemid 26431358)
13. C. Daganzo, M. Cassidy, and R. Bertini, Possible explanations of phase transitions in highway traffic, Transportation Res. Part A, 33 (1999), pp. 365-379. (Pubitemid 29328677)
14. J. Del Castillo and F. Benitez, On the functional form of the speed-density relationship I: General theory, Transportation Res. Part B, 29 (1995), pp. 373-389.
15. J. Del Castillo, P. Pintado, and F. Benitez, The reaction time of drivers and the stability of traffic flow, Transportation Res. Part B, 28 (1994), pp. 35-60.
16. M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Ser. Appl. Math. 1, AMS, Springfield, MO, 2006.
17. M. Garavello and B. Piccoli, On fluido-dynamic models for urban traffic, Netw. Heterog. Media, 4 (2009), pp. 107-126.
18. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), pp. 697-715.
19. P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), pp. 287-303. (Pubitemid 43851204)
20. S. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Sb. Math., 89 (1959), pp. 271-306.
21. H. Greenberg, An analysis of traffic flow, Oper. Res., 7 (1959), pp. 79-85.
22. B. Greenshields, A study of traffic capacity, Proceedings of the Highway Research Board, 14 (1935), pp. 448-477.
23. H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Appl. Math. Sci. 152, Springer-Verlag, New York, 2002.
24. Z. Jia, C. Chen, B. Coifman, and P. Varaiya, The PeMS algorithms for accurate, real-time estimates of g-factors and speeds from single-loop detectors, in Intelligent Transportation Systems, 2001, pp. 536-541. (Pubitemid 32936397)
25. B. Kerner, Experimental features of self-organization in traffic flow, Phys. Rev. Lett., 81 (1998), pp. 3797-3800. (Pubitemid 128629616)
26. B. Kerner, Phase transitions in traffic flow, Traffic and Granular Flow, (2000), pp. 253-283.
27. S. Kruzhkov, First order quasilinear equations in several space variables, Sb. Math., 10 (1970), pp. 217-243.
28. J. Lebacque, The Godunov scheme and what it means for first order macroscopic traffic flow models, in Proceedings of the 13th ISTTT, Lyon, France, 1996, pp. 647-677.
29. P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary conditions, Math. Methods Appl. Sci., 10 (1988), pp. 265-287.
30. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, UK, 2002.
31. M. Lighthill and G. Whitham, On kinematic waves II: A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 229 (1956), pp. 317-345.
32. W. Lin and D. Ahanotu, Validating the Basic Cell Transmission Model on a Single Freeway Link, Technical report, Institute of Transportation Studies, University of California, Berkeley, 1995.
33. L. Munoz, X. Sun, R. Horowitz, and L. Alvarez, Traffic density estimation with the cell transmission model, in Proceedings of the 2003 American Control Conference, Vol. 5, 2003, pp. 3750-3755.
34. G. Newell, A simplified theory of kinematic waves in highway traffic II: Queueing at freeway bottlenecks, Transportation Res. Part B, 27 (1993), pp. 289-303.
35. O. Oleinik, Discontinuous solutions of non-linear differential equations, Uspekhi Mat. Nauk, 12 (1957), pp. 3-73.
36. M. Papageorgiou, Some remarks on macroscopic traffic flow modelling, Transportation Res. Part A, 32 (1998), pp. 323-329. (Pubitemid 128714046)
37. H. Payne, Models of freeway traffic and control, Math. Models Public Syst., 1 (1971), pp. 51-61.
38. P. Richards, Shock waves on the highway, Oper. Res., 4 (1956), pp. 42-51.
39. D. Serre, Systems of Conservation Laws, Diderot, Paris, 1996.
40. I. Strub and A. Bayen, Weak formulation of boundary conditions for scalar conservation laws: An application to highway traffic modelling, Internat. J. Robust Nonlinear Control, 16 (2006), pp. 733-748. (Pubitemid 44683306)
41. B. Temple, Systems of conversation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), pp. 781-795.
42. L. Tong, Nonlinear dynamics of traffic jams, Phys. D, 207 (2005), pp. 41-51.
43. E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, 1997.
44. R. Underwood, Speed, volume, and density relationships: Quality and theory of traffic flow, Yale Bureau of Highway Traffic, (1961), pp. 141-188.
45. P. Varaiya, Reducing highway congestion: An empirical approach, Eur. J. Control, 11 (2005), pp. 301-309.
46. P. Varaiya, Congestion, ramp metering and tolls, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366 (2008), pp. 1921-1930. (Pubitemid 351585436)
47. Y. Wang and M. Papageorgiou, Real-time freeway traffic state estimation based on extended Kalman filter: A general approach, Transportation Res. Part B, 39 (2005), pp. 141-167. (Pubitemid 40032216)
48. G. Whitham, Linear and Nonlinear Waves, Pure Appl. Math., Wiley Interscience, New York, 1974.
49. H. Zhang, A theory of nonequilibrium traffic flow, Transportation Res. Part B, 32 (1998), pp. 485-498. (Pubitemid 128407015)
50. H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. Part B, 36 (2002), pp. 275-290. (Pubitemid 34043312)