The European Physical Journal B

, Volume 69, Issue 4, pp 549–562 | Cite as

On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models

Interdisciplinary Physics

Abstract

Daganzo’s criticisms of second-order fluid approximations of traffic flow [C. Daganzo, Transpn. Res. B. 29, 277 (1995)] and Aw and Rascle’s proposal how to overcome them [A. Aw, M. Rascle, SIAM J. Appl. Math. 60, 916 (2000)] have stimulated an intensive scientific activity in the field of traffic modeling. Here, we will revisit their arguments and the interpretations behind them. We will start by analyzing the linear stability of traffic models, which is a widely established approach to study the ability of traffic models to describe emergent traffic jams. Besides deriving a collection of useful formulas for stability analyses, the main attention is put on the characteristic speeds, which are related to the group velocities of the linearized model equations. Most macroscopic traffic models with a dynamic velocity equation appear to predict two characteristic speeds, one of which is faster than the average velocity. This has been claimed to constitute a theoretical inconsistency. We will carefully discuss arguments for and against this view. In particular, we will shed some new light on the problem by comparing Payne’s macroscopic traffic model with the Aw-Rascle model and macroscopic with microscopic traffic models.

PACS

89.40.Bb Land transportation 45.70.Vn Granular models of complex systems; traffic flow 83.60.Wc Flow instabilities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Chowdhury, L. Santen, A. Schadschneider, Physics Reports 329, 199 (2000) Google Scholar
  2. D. Helbing, Rev. Modern Phys. 73, 1067 (2001)Google Scholar
  3. T. Nagatani, Reports on Progress in Physics 65, 1331 (2002)Google Scholar
  4. K. Nagel, Multi-Agent Transportation Simulations, see http://www2.tu-berlin.de/fb10/ISS/FG4/archive/sim-archive/publications/book/ Google Scholar
  5. M. Schönhof, D. Helbing, Transportation Science 41, 135 (2007) Google Scholar
  6. Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S.-I. Tadaki, S. Yukawa, New J. Phys. 10, 033001 (2008) Google Scholar
  7. R. Herman, E.W. Montroll, R.B. Potts, R.W. Rothery, Operations Research 7, 86 (1959) Google Scholar
  8. R.D. Kühne, M.B. Rödiger, In Proceedings of the 1991 Winter Simulation Conference, edited by B.L. Nelson, W.D. Kelton, G.M. Clark (Society for Computer Simulation International, Phoenix, AZ, 1991), pp. 762–770 Google Scholar
  9. M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Phys. Rev. E 51, 1035 (1995) Google Scholar
  10. D. Helbing, Eur. Phys. J. B, in press (2009), see e-print http://arxiv.org/abs/0805.3400. Google Scholar
  11. C.F. Daganzo, Transp. Res. B 29, 277 (1995) Google Scholar
  12. M. Lighthill, G. Whitham, Proc. Roy. Soc. of London A 229, 317 (1955) Google Scholar
  13. P.I. Richards, Operations Research 4, 42 (1956) Google Scholar
  14. G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974) Google Scholar
  15. H.J. Payne, In Mathematical Models of Public Systems, edited by G.A. Bekey (Simulation Council, La Jolla, CA, 1971), Vol. 1, pp. 51–61 Google Scholar
  16. H.J. Payne, In Research Directions in Computer Control of Urban Traffic Systems, edited by W.S. Levine, E. Lieberman, J.J. Fearnsides (American Society of Civil Engineers, New York, 1979), pp. 251–265 Google Scholar
  17. I. Prigogine, R. Herman, Kinetic Theory of Vehicular Traffic (Elsevier, New York, 1971) Google Scholar
  18. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 59, 239 (1999) Google Scholar
  19. V. Shvetsov, D. Helbing, Phys. Rev. E 59, 6328 (1999) Google Scholar
  20. D. Helbing, M. Treiber, Computing in Science & Engineering 1, 89 (1999) Google Scholar
  21. C.K.J. Wagner, Verkehrsflußmodelle unter Berücksichtig-ung eines internen Freiheitsgrades, Ph.D. thesis, TU Munich, 1997 Google Scholar
  22. S.P. Hoogendoorn, P.H.L. Bovy, Transp. Res. B 34, 123 (2000) Google Scholar
  23. S.L. Paveri-Fontana, Transportation Research 9, 225 (1975) Google Scholar
  24. L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, Rhode Island, 1998) Google Scholar
  25. S.F. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1993) Google Scholar
  26. R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992) Google Scholar
  27. A. Aw, M. Rascle, SIAM J. Appl. Math. 60, 916 (2000) Google Scholar
  28. A. Klar, R. Wegener, SIAM J. Appl. Math. 60, 1749 (2000) Google Scholar
  29. J.M. Greenberg, SIAM J. Appl. Math. 62, 729 (2001) Google Scholar
  30. H.M. Zhang, Transp. Res. B 36, 275 (2002) Google Scholar
  31. P. Goatin, Math. Comp. Modelling 44, 287(2006) Google Scholar
  32. M. Garavello, B. Piccoli, Communications in Partial Differential Equations 31, 243 (2006) Google Scholar
  33. F. Siebel, W. Mauser, Phys. Rev. E 73, 066108 (2006) Google Scholar
  34. Z.-H. Ou, S.-Q. Dai, P. Zhang, L.-Y. Dong, SIAM J. Appl. Math. 67, 605 (2007) Google Scholar
  35. J.-P. Lebacque, S. Mammar, H. Haj-Salem, Trans. Res. B 41, 710 (2007) Google Scholar
  36. F. Berthelin, P. Degond, M. Delitala, M. Rascle, Arch. Rational Mech. Anal. 187, 185 (2008) Google Scholar
  37. J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987) Google Scholar
  38. W.F. Phillips, Transportation Planning and Technology 5, 131 (1979) Google Scholar
  39. B.S. Kerner, P. Konhäuser, Phys. Rev. E 48, R2335 (1993) Google Scholar
  40. H.Y. Lee, H.-W. Lee, D. Kim, Phys. Rev. Lett. 81, 1130 (1998) Google Scholar
  41. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62, 1805 (2000) Google Scholar
  42. A. Hood, Characteristics, in Encyklopedia of Nonlinear Science, edited by A. Scott (Routledge, New York, 2005) Google Scholar
  43. D. Helbing, Phys. A 233, 253 (1996), see also http://arxiv.org/abs/cond-mat/9805136 Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.ETH Zurich, UNO D11, Universitätstr. 41ZurichSwitzerland

Personalised recommendations