The Riemann problem for the Aw-Rascle model in the traffic flow with the Delta initial data for the Chaplygin gas is studied. The solutions are constructed globally under the generalized Rankine-Hugoniot relations, the δ-entropy conditions, and the gen- eralized δ-entropy conditions. A new Delta wave, which is called a primary Delta wave, is defined in some solutions. The primary Delta wave satisfies the generalized Rankine- Hugoniot relations and the generalized δ-entropy conditions. It generates initially from the Delta initial data, which either evaluates a Delta wave, whose weight becomes stronger and stronger, or disappears at a finite time.
Wancheng SHENG;Ying ZENG
. Generalized δ-entropy condition to Riemann solution for Chaplygin gas in traffic flow[J]. Applied Mathematics and Mechanics, 2015
, 36(3)
: 353
-364
.
DOI: 10.1007/s10483-015-1915-6
[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 A: Mathematical, Physical and Engineering Sci- ences, 229A, 317-345 (1955)
[2] Richards, P. Shock waves on the highway. Operations Research, 4, 42-51 (1956)
[3] Payne, H. J. Models of Freeway Traffic and Control, Simulation Councils, San Diego, 51-61 (1971)
[4] Whitham, G. B. Linear and Nonlinear Waves, Wiley InterScience, New York, 75-102 (1974)
[5] Aw, A. and Rascle, M. Resurrection of second order models of traffic flow. Journal on Applied Mathematics, 60, 916-938 (2000)
[6] Daganzo, C. F. Requiem for second-order fluid approximation of traffic flow. Transportation Research: Part B, 29, 277-286 (1995)
[7] Colombo, R. M. Hyperbolic conservation laws in traffic flow. Journal on Applied Mathematics, 63, 708-721 (2002)
[8] Sun, M. N. Interactions of elementary waves for the AW-Rascle model. Journal on Applied Math- ematics, 69, 1542-1558 (2009)
[9] Sheng, W. C. and Zhang, T. The Riemann problem for the transportation equations in gas dynamics. Memoirs of the American Mathematical Society, 137, 1-77 (1999)
[10] Li, J. Q., Zhang, T., and Yang, S. L. The Two-Dimensional Riemann Problem in Gas Dynamics, Chapman and Hall/CRC, New York, 38-66 (1998)
[11] Guo, L. H., Sheng, W. C., and Zhang, T. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system. Communications on Pure and Applied Analysis, 9, 431-458 (2010)
[12] Wang, Z. and Zhang, Q. L. The Riemann problem with Delta initial data for the one-dimensional Chaplygin gas equations. Acta Mathematica Scientia, 32, 825-841 (2012)
[13] Pan, L. J. and Han, X. L. The Aw-Rascle traffic model with Chaplygin pressure. Journal of Mathematical Analysis and Applications, 401, 379-387 (2013)
[14] Li, J. Y. The Riemann problem with the Delta initial data for the Chaplygin pressure Aw-Rascle traffic model (in Chinese). Advances in Apllied Mathematics, 2, 114-126 (2013)
[15] Chang, T. and Hsiao, L. The Riemann Problem and Interaction of Waves in Gas Dynamics, John Wiley and Sons, New York, 95-161 (1989)
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