Advertisement

A Characteristic Particle Method for Traffic Flow Simulations on Highway Networks

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 89)

Abstract

A characteristic particle method for the simulation of first order macroscopic traffic models on road networks is presented. The approach is based on the method particleclaw, which solves scalar one dimensional hyperbolic conservation laws exactly, except for a small error right around shocks. The method is generalized to nonlinear network flows, where particle approximations on the edges are suitably coupled together at the network nodes. It is demonstrated in numerical examples that the resulting particle method can approximate traffic jams accurately, while only devoting a few degrees of freedom to each edge of the network.

Keywords

Particle Characteristic Particleclaw Shock Traffic flow Network 

Notes

Acknowledgements

Y. Farjoun was partially financed by the Spanish Ministry of Science and Innovation under grant FIS2008-04921-C02-01. The authors would like to acknowledge support by the National Science Foundation. B. Seibold was supported through grant DMS–1007899. In addition, B. Seibold wishes to acknowledge partial support by the National Science Foundation through grants DMS–0813648 and DMS–1115278. Y. Farjoun wishes to acknowledge partial support through grant DMS–0703937. In addition, Farjoun thanks Luis Bonilla at the UC3M and Rodolfo R. Rosales at MIT for providing a framework under which this work was done.

References

  1. 1.
    D. Armbruster, D. Marthaler, C. Ringhofer, Kinetic and fluid model hierarchies for supply chains. Multiscale Model. Simul. 2, 43–61 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A. Aw, M. Rascle, Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60, 916–944 (2000)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M.J. Berger, J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    G. Bretti, R. Natalini, B. Piccoli, Fast algorithms for the approximation of a traffic flow model on networks. Discret. Contin. Dyn. Syst. Ser. B 6, 427–448 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    G. Bretti, R. Natalini, B. Piccoli, Numerical approximations of a traffic flow model on networks. Netw. Heterog. Media 1, 57–84 (2006)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    B. Cockburn, C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.M. Coclite, M. Garavello, B. Piccoli, Traffic flow on a road network. SIAM J. Math. Anal. 36, 1862–1886 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    R. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63, 708–721 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  10. 10.
    Y. Farjoun, B. Seibold, Solving one dimensional scalar conservation laws by particle management, in Meshfree Methods for Partial Differential Equations IV, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Computational Science and Engineering, vol. 65 (Springer, Berlin, 2008), pp. 95–109Google Scholar
  11. 11.
    Y. Farjoun, B. Seibold, An exactly conservative particle method for one dimensional scalar conservation laws. J. Comput. Phys. 228, 5298–5315 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Y. Farjoun, B. Seibold, A rarefaction-tracking method for conservation laws. J. Eng. Math. 66, 237–251 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Y. Farjoun, B. Seibold, An exact particle method for scalar conservation laws and its application to stiff reaction kinetics, in Meshfree Methods for Partial Differential Equations V, M. Griebel, M.A. Schweitzer. Lecture Notes in Computational Science and Engineering, vol. 79 (Springer, Heidelberg/Berlin, 2011), pp. 105–124Google Scholar
  14. 14.
    M.R. Flynn, A.R. Kasimov, J.-C. Nave, R.R. Rosales, B. Seibold, Self-sustained nonlinear waves in traffic flow. Phys. Rev. E 79, 056113 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    S.K. Godunov, A difference scheme for the numerical computation of a discontinuous solution of the hydrodynamic equations. Math. Sb. 47, 271–306 (1959)MathSciNetGoogle Scholar
  16. 16.
    B.D. Greenshields, A study of traffic capacity. Proc. Highw. Res. Rec. 14, 448–477 (1935)Google Scholar
  17. 17.
    A. Harten, B. Engquist, S. Osher, S. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    D. Helbing, Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)CrossRefGoogle Scholar
  19. 19.
    M. Herty, A. Klar, Modelling, simulation and optimization of traffic flow networks. SIAM J. Sci. Comput. 25, 1066–1087 (2003)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    M. Herty, A. Klar, B. Piccoli, Existence of solutions for supply chain models based on partial differential equations. SIAM J. Math. Anal. 39, 160–173 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    H. Holden, L. Holden, R. Hegh-Krohn, A numerical method for first order nonlinear scalar conservation laws in one dimension. Comput. Math. Appl. 15, 595–602 (1988)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    H. Holden, N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26, 999–1017 (1995)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    H. Holden, N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws (Springer, New York, 2002)MATHGoogle Scholar
  24. 24.
    P.D. Lax, B. Wendroff, Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    M.J. Lighthill, G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of Royal Society A, Piccadilly, London, vol. 229 (1955), pp. 317–345MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    H.J. Payne, FREEFLO: A macroscopic simulation model of freeway traffic. Transp. Res. Rec. 722, 68–77 (1979)Google Scholar
  28. 28.
    W.H. Reed, T.R. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory (1973)Google Scholar
  29. 29.
    P.I. Richards, Shock waves on the highway, Oper. Res. 4, 42–51 (1956)MathSciNetCrossRefGoogle Scholar
  30. 30.
  31. 31.
    B. van Leer, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–370 (1974)MATHGoogle Scholar
  32. 32.
    H.M. Zhang, A theory of non-equilibrium traffic flow. Transp. Res. B 32, 485–498 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Broad Institute of MIT and HarvardCambridgeUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations