Discrete-velocity models and relaxation schemes for traffic flows

SIAM Journal on Scientific Computing

Volumn 28, Issue 4, 2006, Pages 1582-1596

  Herty, Michael ^a   Pareschi, Lorenzo ^b   Seaid, Mohammed ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF FERRARA   (Italy)

Author keywords

Discrete velocity models; ENO schemes; IMEX schemes; Lighthill Whitham Richards equation; Relaxation schemes

Indexed keywords

MATHEMATICAL MODELS; NUMERICAL METHODS; RELAXATION PROCESSES; VELOCITY CONTROL;

DISCRETE MODELS; LIGHTHILL-WHITHAM-RICHARDS EQUATIONS; TRAFFIC FLOWS; TRAFFIC JAMS;

TRAFFIC CONTROL;

EID: 34547230580     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/04061982X     Document Type: Article

References (27)

1. A. AW AND M. RASCLE, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916-938.
2. G. Q. CHEN, C. D. LEVERMORE, AND T. P. LIU, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 787-830.
3. C. F. DAGANZO, Requiem for second order fluid approximations of traffic flow, Transportation Res. Part B., 29 (1995), pp. 277-286.
4. C. F. DAGANZO, A behavioral theory of multi-lane traffic flow. Part I. Long homogenous freeway sections, Transportation Res. Part B., 36 (2002), pp. 131-158.
5. S. GOTTLIEB, C-W. SHU, AND E. TADMOR, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89-112.
6. G.-S. JIANG, D. LEVY, C.-T. LIN, S. OSHER, AND E. TADMOR, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2147-2168.
7. S. JIN, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 122 (1995), pp. 51-67.
8. S. JIN AND Z. XIN, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 235-277.
9. A. KLAR, Relaxation schemes for a lattice-Boltzmann-type discrete velocity model and numerical Navier-Stokes limit, J. Comput. Phys., 148 (1999), pp. 416-432.
10. X.-G. LI, X.-J. YU, AND G.-N. CHEN, The third-order relaxation schemes for hyperbolic conservation laws, J. Comput. Appl. Math., 138 (2002), pp. 93-108.
11. X. LIU AND S. OSHER, Convex ENO high order multi-dimensonal schemes without field by field decomposition or staggered grids, J. Comput. Phys., 141 (1998), pp. 1-27.
12. M. J. LIGHTHILL AND J. B. WHITHAM, On kinematic waves, Proc. R. Soc. Lond. Ser. A, 229 (1955), pp. 281-345.
13. A. MAY, Traffic Flow Fundamentals, Prentice-Hall, Englewood Cliffs, NJ, 1990.
14. R. NATALINI, Convergence to equilibrium for relaxation approximations of conservation laws, Comm. Pure Appl. Math., 49 (1996), pp. 795-823.
15. R. NATALINI, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations, 148 (1998), pp. 292-317.
16. L. PARESCHI AND G. RUSSO, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, J. Sci. Comput., 25 (2005), pp. 129-155.
17. L. PARESCHI AND M. SEAÏD, A new Monte Carlo approach for conservation laws and relaxation systems, in Proceedings of the 4th International Conference on Computational Science, Lecture Notes in Comput. Sci. 3037, Springer, Berlin, 2004, pp. 276-283.
18. S. L. PAVERI-FONTANA, On Boltzmann-like treatments for traffic flow, Transportation Res., 9 (1975), p. 225.
19. H. PAYNE, FREFLO: A macroscopic simulation model of freeway traffic, Transportation Res. Record, 722 (1979), pp. 68-77.
20. M. SEAÏD, Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions, Internat. J. Numer. Methods Fluids, 46 (2004), pp. 457-484.
21. C-W. SHU, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, A. Quarteroni, ed., Lecture Notes in Math. 1697, Springer, 1998, pp. 325-432.
22. C. W. SHU AND S. OSHER, Efficient implementation of essential non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
23. F. SIEBEL AND W. MAUSER, On the fundamental diagram of traffic flow, SIAM J. Appl. Math., 66 (2006), pp. 1150-1162.
24. P. ZHANG AND R. LIU, Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions, Numer. Methods Partial Differential Equations, 21 (2004), pp. 80-88.
25. P. ZHANG AND R. LIU, Hyperbolic conservation laws with space dependent flux. I. Characteristics theory and Riemann problem, J. Comput. Appl. Math., 156 (2003), pp. 1-21.
26. P. ZHANG AND R. LIU, Hyperbolic conservation laws with space dependent fluxes. II. General study on numerical fluxes, J. Comput. Appl. Math., 176 (2005), pp. 105-129.
27. P. ZHANG, R. LIU, AND S. C WONG, High-resolution numerical approximation of traffic flow problems with variable lanes and free flow velocities, Phys. Rev. E, 71 (2005), 56704.