Skip to main content
SpringerLink
Log in

A Model for the Formation and Evolution of Traffic Jams

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper, we establish and analyze a traffic flow model which describes the formation and dynamics of traffic jams. It consists of a pressureless gas dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. From this analysis, we deduce the particular dynamical behavior of clusters (or traffic jams), defined as intervals where the density limit is reached. An existence result for a generic class of initial data is proved by means of an approximation of the solution by a sequence of clusters. Finally, numerical simulations are produced.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aw A., Klar A., Materne A., Rascle M. (2002) Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J. Appl. Math. 63:259–278

    Article  MATH  MathSciNet  Google Scholar 

  2. Aw A., Rascle M. (2000) Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60:916–938

    Article  MATH  MathSciNet  Google Scholar 

  3. Berthelin F. (2002) Existence and weak stability for a two-phase model with unilateral constraint. Math. Models Methods Appl. Sci. 12:249–272

    Article  MATH  MathSciNet  Google Scholar 

  4. Berthelin F., Bouchut F. (2003) Weak solutions for a hyperbolic systems with unilateral constraint and mass loss. Ann. Inst. H. Poincare Anal. Non Linéaire 20:975–997

    Article  MATH  MathSciNet  Google Scholar 

  5. Bouchut, F.: On zero pressure gas dynamics. In: (Ed. Perthame, B.), Advances in kinetic theory and computing, World Scientific (1994)

  6. Bouchut F., Brenier Y., Cortes J., Ripoll J.F. (2000) A hierachy of models for two-phase flows. J. Nonlinear Sci. 10:639–660

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bouchut F., James F. (1999) Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differential Equations 24:2173–2189

    Article  MATH  MathSciNet  Google Scholar 

  8. Brenier Y., Grenier E. (1998) Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35:2317–2328

    Article  MATH  MathSciNet  Google Scholar 

  9. Colombo R.M. (2002) A 2 × 2 hyperbolic traffic flow model Traffic Hav modelling and simulation. Math. Comp. Modelling 35:683–688

    Article  MATH  MathSciNet  Google Scholar 

  10. Dafermos, C.: Hyperbolic conservation laws in continuum physics. Springer-Verlag, 1999

  11. Daganzo C. (1995) Requiem for second order fluid approximations of traffic flow. Transp. Res. B 29B:277–286

    Article  Google Scholar 

  12. E. W., Rykov, Y.G., Sinai, Y.G. (1996) Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys. 177, 349–380

  13. Gazis D.C., Herman R., Rothery R. (1961) Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9:545–567

    Article  MATH  MathSciNet  Google Scholar 

  14. Grenier E. (1995) Existence globale pour le système des gaz sans pression. C. R. Math. Acad. Sci. Paris 321:171–174

    MATH  MathSciNet  Google Scholar 

  15. Klar, A., Wegener, R.: Traffic flow: models and numerics. In: Modeling and Computational Methods for Kinetic Equations, Birkhäuser, Boston, 219–259 (2004)

  16. Klar A., Wegener R. (1997) Enskog-like kinetic models for vehicular traffic. J. Stat. Phys. 87:91–114

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Lighthill, M.J., Whitham, J.B.: On kinematic waves. I: flow movement in long rivers. II: A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Sec A 229, Math. Phys. Eng. Sci. 281–345 (1955)

  18. Nagel K., Schreckenberg M. (1992) A cellular automaton model for freeway traffic. J. Physique 2:2221–2229

    ADS  Google Scholar 

  19. Nelson P. (1995) A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions. Transport. Theory Statist. Phys. 24:383–409

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Paveri-Fontana S.L. (1975) On Boltzmann-like treatments for traffic flow. Transportation Res. 9:225–235

    Article  Google Scholar 

  21. Payne, H.J.: Models of Freeway Traffic and Control. In: Mathematical Models of Public Systems, Simulation Council Proc. 1, (1971)

  22. Payne H.J. (1979) FREFLO: A macroscopic simulation model of freeway traffic. Transportation Research Record 722:68–75

    Google Scholar 

  23. Prigogine I., Herman R (1971) Kinetic theory of vehicular traffic. American Elsevier Publishing Co, New-York

    MATH  Google Scholar 

  24. Serre, D.: Systems of conservation laws 1 & 2. Cambridge University Press, 1999 and 2000

  25. Smoller, J.: Shock waves and reaction-diffusion equations. Springer-Verlag, 1983

  26. Temple B. (1983) Systems of conservation laws with coinciding shock and rarefaction curves. Contemp. Math. 17:143–151

    MATH  Google Scholar 

  27. Zhang M. (2002) A non-equilibrium traffic model devoid of gas-like behavior. Transportation Res. B 36:275–290

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire J. A. Dieudonné, Université de Nice, Nice, France

    F. Berthelin & M. Rascle

  2. MIP UMR 5640, Université Paul Sabatier, Toulouse, France

    P. Degond

  3. Department of Mathematics, Politecnico di Torino, Torino, Italy

    M. Delitala

Authors
  1. F. Berthelin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Degond
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Delitala
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Rascle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Berthelin.

Additional information

Communicated by C. M. Dafermos

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berthelin, F., Degond, P., Delitala, M. et al. A Model for the Formation and Evolution of Traffic Jams. Arch Rational Mech Anal 187, 185–220 (2008). https://doi.org/10.1007/s00205-007-0061-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-007-0061-9

Keywords

Search

Navigation