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.
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References
D. Chowdhury, L. Santen, A. Schadschneider, Physics Reports 329, 199 (2000)
D. Helbing, Rev. Modern Phys. 73, 1067 (2001)
T. Nagatani, Reports on Progress in Physics 65, 1331 (2002)
K. Nagel, Multi-Agent Transportation Simulations, see http://www2.tu-berlin.de/fb10/ISS/FG4/archive/sim-archive/publications/book/
M. Schönhof, D. Helbing, Transportation Science 41, 135 (2007)
Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, S.-I. Tadaki, S. Yukawa, New J. Phys. 10, 033001 (2008)
R. Herman, E.W. Montroll, R.B. Potts, R.W. Rothery, Operations Research 7, 86 (1959)
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
M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Phys. Rev. E 51, 1035 (1995)
D. Helbing, Eur. Phys. J. B, in press (2009), see e-print http://arxiv.org/abs/0805.3400.
C.F. Daganzo, Transp. Res. B 29, 277 (1995)
M. Lighthill, G. Whitham, Proc. Roy. Soc. of London A 229, 317 (1955)
P.I. Richards, Operations Research 4, 42 (1956)
G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)
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
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
I. Prigogine, R. Herman, Kinetic Theory of Vehicular Traffic (Elsevier, New York, 1971)
M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 59, 239 (1999)
V. Shvetsov, D. Helbing, Phys. Rev. E 59, 6328 (1999)
D. Helbing, M. Treiber, Computing in Science & Engineering 1, 89 (1999)
C.K.J. Wagner, Verkehrsflußmodelle unter Berücksichtig-ung eines internen Freiheitsgrades, Ph.D. thesis, TU Munich, 1997
S.P. Hoogendoorn, P.H.L. Bovy, Transp. Res. B 34, 123 (2000)
S.L. Paveri-Fontana, Transportation Research 9, 225 (1975)
L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence, Rhode Island, 1998)
S.F. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1993)
R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992)
A. Aw, M. Rascle, SIAM J. Appl. Math. 60, 916 (2000)
A. Klar, R. Wegener, SIAM J. Appl. Math. 60, 1749 (2000)
J.M. Greenberg, SIAM J. Appl. Math. 62, 729 (2001)
H.M. Zhang, Transp. Res. B 36, 275 (2002)
P. Goatin, Math. Comp. Modelling 44, 287(2006)
M. Garavello, B. Piccoli, Communications in Partial Differential Equations 31, 243 (2006)
F. Siebel, W. Mauser, Phys. Rev. E 73, 066108 (2006)
Z.-H. Ou, S.-Q. Dai, P. Zhang, L.-Y. Dong, SIAM J. Appl. Math. 67, 605 (2007)
J.-P. Lebacque, S. Mammar, H. Haj-Salem, Trans. Res. B 41, 710 (2007)
F. Berthelin, P. Degond, M. Delitala, M. Rascle, Arch. Rational Mech. Anal. 187, 185 (2008)
J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987)
W.F. Phillips, Transportation Planning and Technology 5, 131 (1979)
B.S. Kerner, P. Konhäuser, Phys. Rev. E 48, R2335 (1993)
H.Y. Lee, H.-W. Lee, D. Kim, Phys. Rev. Lett. 81, 1130 (1998)
M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62, 1805 (2000)
A. Hood, Characteristics, in Encyklopedia of Nonlinear Science, edited by A. Scott (Routledge, New York, 2005)
D. Helbing, Phys. A 233, 253 (1996), see also http://arxiv.org/abs/cond-mat/9805136
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Helbing, D., Johansson, A. On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models. Eur. Phys. J. B 69, 549–562 (2009). https://doi.org/10.1140/epjb/e2009-00182-7
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DOI: https://doi.org/10.1140/epjb/e2009-00182-7