Hyperbolicity and kinematic waves of a class of multi-population partial differential equations

European Journal of Applied Mathematics

Volumn 17, Issue 2, 2006, Pages 171-200

  Zhang, Peng ^a   Liu, Ru Xun ^b   Wong, S C ^c   Dai, Shi Qiang ^a  

^a SHANGHAI UNIVERSITY   (China)

^b UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^c UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33749354424     PISSN: 09567925     EISSN: 14694425     Source Type: Journal    
DOI: 10.1017/S095679250500642X     Document Type: Article

References (24)

1. BAGNERINI, P. & RASCLE, M. (2003) A multiclass homogenized hyperbolic model of traffic flow. SIAM J. Math. Anal. 35, 949-937.
2. BENZONI-GAVAGE, S. & COLOMBO, R. M. (2003) An n-populations model for traffic flow. Euro. J. Appl. Math. 14, 587-612.
3. CHOWDHURY, D., SANTEN, L. & SCHADSCHNEIDER, A. (2000) Statistical physics of vehicular traffic and some related systems. Physics Reports, 329, 199-329.
4. GREENBERG, J. M., KLAR, A. & RASCLE, M. (2003) Congestion on Multi-lane highways. SIAM J. Appl. Math. 63, 818-833.
5. HARTEN, A., LAX, P. D. & VAN LEER, B. (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 1, 35-61.
6. HARTEN, A., ENGQUIST, B., OSHER, S. & CHAKRAVARTHY, S. (1987) Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231-303.
7. HELBING, D. (2001) Traffic and related self-driven many partical systems. Rev. Mod. Phys. 1067-1141.
8. LAX, P. D. (1957) Hyperbolic Systems of Conservation Laws II. Comm. Pure Appl. Math. 10, 537-566.
9. LAX, P. D. (1971) Shock waves and entropy. Contributions to Nonlinear Functional Anal. Proc. Symposium at the University of Wisconsin, 10, 603.
10. LAX, P. D. (1973) Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia.
11. LIGHTHILL, M. J. & WHITHAM, G. B. (1955) On kinematic waves: II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc.London, Ser. A 229, 317-345.
12. LIU, X.-D., OSHER, S. & CHAN, T. (1994) Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115, 200-212.
13. ROE, P. L. (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357-372.
14. RICHARDS, P. I. (1956) Shock waves on the highway. Operat. Res. 4, 42-51.
15. SHU, C.-W. (1956) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Report No. 1997-65.
16. SMOLLER, I. (1994) Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York.
17. SMOLLER, I. (1999) Riemann Solvers and Numerical Methods for Fluid Dynamics. Wiley.
18. WHITHAM, G. B. (1974) Linear and Nonlinear Waves. Springer-Verlay.
19. WONG, G. C. K. & WONG, S. C. (2002) A multi-class traffic flow model-an extension of LWR model with heterogeneous drivers. Transpn. Res. A 36, 827-841.
20. GOTTLIEB, S., SHU, C.-W. & TADMOR, E. (2001) Strong stability preserving high order time discretization method. SIAM Rev. 43, 89-112.
21. ZHANG, M. P., SHU, C.-W., WONG, G. C. K. & WONG, S. C. (2003) A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model. J. Comput. Phys. 191, 639-659.
22. ZHANG, P. & LIU, R. X. (2003) Hyperbolic conservation laws with space-dependent flux: I. Characteristics theory and Riemann problem. J. Comput. Appl. Math. 156, 1-21.
23. ZHANG, P. & LIU, R. X. (2005) Hyperbolic conservation laws with space-dependent flux: II. General study of numerical fluxes. J. Comput. Appl. Math. 176, 105-129.
24. ZHANG, P. & LIU, R. X. (2005) Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions. Numer. Meth. Partial Diff. Eq. 21(1), 80-88.