Effects of Relaxation and Anticipation on Riemann Solutions of Payne-Whitham Model
Abstract
The solutions of Riemann problems of a particular higher-order model—the Payne-Whitham (PW) model—are studied using Roe’s flux splitting scheme as presented by Leo and Pretty. Despite numerous works on higher-order models, little is known about Riemann solutions of these models and how relaxation and anticipation affect these solutions. Riemann solutions of the PW model are computed and compared with those of the Lighthill-Whitham-Richards (LWR) model having the same initial (density) data. It was found that faster relaxation forces the PW model to behave much like the LWR model, that strong anticipation has a stabilizing effect on traffic, and that shock waves travel at different speeds in the PW model than they do in the LWR model. These findings provide a basic checklist for experimental validation of PW-like higher models.
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References
1. Lighthill M. J., and Whitham G. B. On Kinematic Waves II: A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society, Vol. 229, No. 1178, 1955, pp. 317–345.
2. Richards P. I. Shock Waves on the Highway. Operational Research, Vol. 4, 1956, pp. 42–51.
3. Payne H. J. Models of Freeway Traffic and Control. In Mathematical Models of Public Systems (Bekey G.A., ed.), Simulation Councils Proceedings Series, Vol. 1, 1971.
4. Whitham G. B. Linear and Nonlinear Waves. John Wiley & Sons, New York, 1974.
5. Papageorgiou M. Applications of Automatic Control Concepts to Traffic Flow Modeling and Control. Lecture Notes in Control and Information Sciences (Balakrishnan A.V. and Thoma M., eds.), Springer-Verlag, 1983.
6. Papageorgiou M., Blosseville J., and Hadj-Salem H. Modeling and Real-Time Control of Traffic Flow on the Southern Part of Boulevard Peripherique in Paris: Part I: Modeling. Transportation Research A, Vol. 24, No. 5, 1990, pp. 345–359.
7. Hauer E., and Hurdle V. F. Discussion. In Payne, H. J., FREFLO: A Macroscopic Simulation Model of Freeway Traffic, Transportation Research Record 722, TRB, National Research Council, Washington, D.C., 1979, pp. 75–76.
8. Derzko N. A., Ugge A. J., and Case E. R. Evaluation of Dynamic Freeway Flow Model by Using Field Data. In Transportation Research Record 905, TRB, National Research Council, Washington, D.C., 1983, pp. 52–60.
9. Daganzo C. F. Requiem for Second-Order Approximations of Traffic Flow. Transportation Research B, Vol. 29, No. 4, 1995, pp. 277–286.
10. Leo C. J., and Pretty R. L. Numerical Simulation of Macroscopic Continuum Traffic Models. Transportation Research B, Vol. 26, No. 3, 1992, pp. 207–220.
11. Zhang H. M. An Analysis of the Stability and Wave Properties of a New Continuum Theory. Transportation Research B, Vol. 33, No. 6, 1999, pp. 387–398.
12. Courant R., and Friedrichs K. O. Supersonic Flow and Shock Waves. Interscience, New York, 1948.
13. Godunov S. K. Mat. Sbornik, Vol. 47, 1959, pp. 271–306.
14. Roe P. L. Approximate Riemann Solvers, Parameter Vectors and Difference Schemes. Journal of Computational Physics, Vol. 43, 1981, pp. 357–372.
15. Bui D. D., Narasimhan S. L., and Nelson P. Computational Realizations of the Entropy Condition in Modeling Congested Traffic Flow. Report FHWA/TX-92/1232-7. FHWA, U.S. Department of Transportation, 1992
16. Daganzo C. F. A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow. Transportation Research B, Vol. 29, No. 4, 1995, pp. 261–276.
17. Lebacque J. P. The Godunov Scheme and What It Means for the First Order Traffic Flow Models. Proc., Thirteenth International Symposium on Transportation and Traffic Theory, 1996.
18. Greenshields B. D. A Study of Traffic Capacity. HRB Proc., Vol. 14, 1934, pp. 448–477.
19. LeVeque R. Numerical Methods for Conservation Laws. Birkhauser Verlag, 1992.
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Article first published: January 2000
Issue published: January 2000
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