Networks and Spatial Economics

, Volume 1, Issue 1–2, pp 167–177 | Cite as

The Mathematical Theory of an Enhanced Nonequilibrium Traffic Flow Model

  • T. Li
  • H. M. ZhangEmail author


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.

non-equilibrium traffic flow global existence large-time behavior Glimm scheme 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Courant, and K.O. Friedrichs, Supersonic Flow and Shock Waves, Springer Verlag, 1948.Google Scholar
  2. 2.
    C. Daganzo, “ Requiem for Second-Order Approximations of Traffic Flow, ” Transportation Research B, 29B, 1995, pp. 277-286.Google Scholar
  3. 3.
    J. Glimm, “ Solutions in the Large for Nonlinear Hyperbolic Systems of Equations, ” Comm. Pure Appl. Math., 18, 1965, pp. 95-105.Google Scholar
  4. 4.
    B. Greenshields, “ A Study of Traffic Capacity, ” Proceedings of the Highway Research Board, 14, 1933, pp. 468-477.Google Scholar
  5. 5.
    B.S. Kerner, and P. Konhauüser, “ Structure and Parameters of Clusters in Traffic Flow, ” Physical Review B, 50(1), 1994, pp. 54-83.Google Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    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.Google Scholar
  8. 8.
    P.D. Lax, “ Hyperbolic Systems of Conservation Laws, II, ” Comm. Pure Appl. Math., 10, 1957, pp. 537-566.Google Scholar
  9. 9.
    T. Li (forthcoming), “Global Solutions and Zero Relaxation limit for a Traffic Flow Model,” SIAM J. Appl. Math., submitted for publication.Google Scholar
  10. 10.
    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.Google Scholar
  11. 11.
    T.-P. Liu, “ The Deterministic Version of the Glimm Scheme, ” Comm. Math. Phys., 57, 1977, pp. 135-148.Google Scholar
  12. 12.
    T.-P. Liu, “ Quasilinear Hyperbolic Systems, ” Comm. Math. Phy., 68, 1979, pp. 141-172.Google Scholar
  13. 13.
    T.-P. Liu, “ Hyperbolic Conservation Laws with Relaxation, ” Comm. Math. Phy., 108, 1987, pp. 153-175.Google Scholar
  14. 14.
    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.Google Scholar
  15. 15.
    P.I. Richards, “ Shock Waves on Highway, ” Operations Research, 4, 1956, pp. 42-51.Google Scholar
  16. 16.
    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.Google Scholar
  17. 17.
    G.B. Whitham, Linear and Nonlinear Waves, New York: Wiley, 1974.Google Scholar
  18. 18.
    H.M. Zhang, “ A Theory of Nonequilibrium Traffic Flow, ” Transportation Research, B, 32(7), 1998, pp. 485-498.Google Scholar
  19. 19.
    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.Google Scholar
  20. 20.
    H.M. Zhang, “ Structural Properties of Solutions Arising from a Nonequilibrium Traffic Flow Theory, ” Transportation Research, B, 34, 2000, pp. 583603.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IowaIowa City
  2. 2.Department of Civil and Environmental EngineeringThe University of CaliforniaDavis

Personalised recommendations