Abstract
This paper establishes the mathematical theory of an enhanced nonequilibrium traffic flow model. The innovation of the model is that it addresses the anisotropic feature of traffic flows. We show rigorously that this new theory reduces to the celebrated LWR theory when the relaxation time goes to zero, that global solutions for this theory exist for initial data of bounded total variation under certain mild conditions, and that the solutions approach the equilibrium solutions exponentially fast. The results justify rigorously the asymptotic equivalence of the relaxation model and the equilibrium equation. Our analysis is based on a generalized Glimm scheme which incorporates the effect of the relaxation source.
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References
R. Courant, and K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer Verlag, 1948.
C. Daganzo, “ Requiem for Second-Order Approximations of Traffic Flow, ” Transportation Research B, 29B, 1995, pp. 277-286.
J. Glimm, “ Solutions in the Large for Nonlinear Hyperbolic Systems of Equations, ” Comm. Pure Appl. Math., 18, 1965, pp. 95-105.
B. Greenshields, “ A Study of Traffic Capacity, ” Proceedings of the Highway Research Board, 14, 1933, pp. 468-477.
B.S. Kerner, and P. Konhauüser, “ Structure and Parameters of Clusters in Traffic Flow, ” Physical Review B, 50(1), 1994, pp. 54-83.
R.D. Kühne, “ Freeway Control and Incident Detection using a Stochastic Continuum Theory of Traffic Flow, ” Proc. 1st Int. Conf. on Applied Advanced Technology in Transportation Engineering, San Diego: CA, 1989, pp. 287-292.
R.D. Kühne, and R. Beckschulte, “ Non-linearity Stochastics of Unstable Traffic Flow. ” In C.F. Daganzo (ed.), Transportation and Traffic Theory, Elsevier Science Publishers, 1993, pp. 367-386.
P.D. Lax, “ Hyperbolic Systems of Conservation Laws, II, ” Comm. Pure Appl. Math., 10, 1957, pp. 537-566.
T. Li (forthcoming), “Global Solutions and Zero Relaxation limit for a Traffic Flow Model,” SIAM J. Appl. Math., submitted for publication.
M.J. Lighthill, and G.B. Whitham, “ On Kinematic Waves: II. A Theory of Traffic Flow on Long Crowded Roads, ” Proc. Roy. Soc., London, Ser. A, 229, 1955, pp. 317-345.
T.-P. Liu, “ The Deterministic Version of the Glimm Scheme, ” Comm. Math. Phys., 57, 1977, pp. 135-148.
T.-P. Liu, “ Quasilinear Hyperbolic Systems, ” Comm. Math. Phy., 68, 1979, pp. 141-172.
T.-P. Liu, “ Hyperbolic Conservation Laws with Relaxation, ” Comm. Math. Phy., 108, 1987, pp. 153-175.
H.J. Payne, “ Models of Freeway Traffic and Control, ” In G.A. Bekey (ed.), Simulation Councils Proc. Ser.: Mathematical Models of Public Systems, Vol. 1, No. 1, 1971.
P.I. Richards, “ Shock Waves on Highway, ” Operations Research, 4, 1956, pp. 42-51.
A. Skabardonis, K. Petty, H. Noeimi, D. Rydzewski, and P. Varaiya, “ I-880 Field Experiment: Data-base Development and Incident Delay Estimation Procedures, ” Transportation Research Record, 1554, 1996, pp. 204-212.
G.B. Whitham, Linear and Nonlinear Waves, New York: Wiley, 1974.
H.M. Zhang, “ A Theory of Nonequilibrium Traffic Flow, ” Transportation Research, B, 32(7), 1998, pp. 485-498.
H.M. Zhang, “ Analyses of the Stability and Wave Properties of a New Continuum Traffic Theory, ” Transportation Research, B, 33(6), 1999, pp. 399-415.
H.M. Zhang, “ Structural Properties of Solutions Arising from a Nonequilibrium Traffic Flow Theory, ” Transportation Research, B, 34, 2000, pp. 583603.
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Li, T., Zhang, H.M. The Mathematical Theory of an Enhanced Nonequilibrium Traffic Flow Model. Networks and Spatial Economics 1, 167–177 (2001). https://doi.org/10.1023/A:1011585212670
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DOI: https://doi.org/10.1023/A:1011585212670