Kinetic Description of Bottleneck Effects in Traffic Flow
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摘要: 采用一个推广的LWR模型研究交通瓶颈效应.通过求解流通量间断的Riemann问题,得到关于模型解结构的解析结果,由此导出了描述在瓶颈上游车流的排队现象及其队列长度和高度(密度)的一个典型解,并能够构造模型方程的一种δ-映射算法.更有意义的是,表明了通过采用三角形基本图,这一运动学模型能够描述时走时停波.通过数值模拟,验证了数值结果与解析结果的一致性,从而支撑了文章的理论结果.Abstract: The effects of traffic bottlenecks using an extended LWR model are dealt with.The solution structure was analytically indicated by study of the Riemann problem,which is characterized by a discontinuo us flux.This leads to a typical solution that describes a queueupstream of the bottleneck and its width and height,and informs the design of a D-mapping algorithm.More significantly,it was found that the kinetic model is able to reproduce stop-and-go waves for a triangular fundamental diagram.Some simulation examples were given to support these conclusions,and are shown to be in agreement with the analytical solutions.
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Key words:
- LWR model /
- discontinuous flux /
- D-mapping algorithm /
- stop-and-gowaves
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[1] Lighthill M J, Whitham G B. On kinematic waves—Ⅱ:a theory of traffic flow on long crowded roads[J].Proceedings of the Royal Society of London, Series A,1955,229(1178):317-345. doi: 10.1098/rspa.1955.0089 [2] Richards P I. Shockwaves on the highway[J].Operations Research,1956,4(1):42-51. doi: 10.1287/opre.4.1.42 [3] ZHANG Peng, LIU Ru-xun. Hyperbolic conservation laws with space-dependent flux—Ⅰ:characteristics theory and Riemann problem[J].Journal of Computational and Applied Mathematics,2003,156(1):1-21. doi: 10.1016/S0377-0427(02)00880-4 [4] ZHANG Peng, LIU Ru-xun. Hyperbolic conservation laws with space-dependent flux—Ⅱ:general study on numerical fluxes[J].Journal of Computational and Applied Mathematics, 2005,176(1):105-129. doi: 10.1016/j.cam.2004.07.005 [5] ZHANG Peng, LIU Ru-xun. Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions[J].Numerical Methods for Partial Differential Equations,2005,21(1):80-88. doi: 10.1002/num.20023 [6] Bürger R , Gracía A, Karlsen K H,et al.A family of numerical schemes for kinematic flows with discontinuous flux[J].Journal of Engineering Mathematics,2008,60(3/4):387-425. doi: 10.1007/s10665-007-9148-4 [7] Bürger R, Gracía A, Karlsen K H,et al.Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[J].Network Heterogeneous Media,2008,3(1):1-41. doi: 10.3934/nhm.2008.3.1 [8] Lin W H, Lo H K. A theoretical probe of a German experiment on stationary moving traffic jams[J].Transportation Research Part B,2003,37(3):251-261. doi: 10.1016/S0191-2615(02)00012-7 [9] Kerner B S, Konhuser P. Structure and parameters of clusters in traffic flow[J]. Physical Review E,1994,50(1):54-83. doi: 10.1103/PhysRevE.50.54 [10] Greenberg J M. Congestion redux[J].SIAM Journal on Applied Mathematics,2004,64(4):1175-1185. doi: 10.1137/S0036139903431737 [11] Siebel F, Mauser W. On the fundamental diagram of traffic flow[J].SIAM Journal on Applied Mathematics,2006,66(4):1150-1162. doi: 10.1137/050627113 [12] Siebel F, Mauser W. Synchronized flow and wide moving jams from balanced vehicular traffic[J].Physical Review E,2006,73(6):066108. doi: 10.1103/PhysRevE.73.066108 [13] Siebel F, Mauser W, Moutari S,et al. Balanced vehicular traffic at a bottleneck[J].Mathematical and Computer Modelling,2009,49(3/4): 689-702. doi: 10.1016/j.mcm.2008.01.006 [14] Zhang P, Wong S C. Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models[J].Physical Review E,2006,74(2):026109. doi: 10.1103/PhysRevE.74.026109 [15] Xu R Y, Zhang P, Dai S Q,et al. Admissibility of a wide cluster solution in anisotropic higher-order traffic flow models[J].SIAM Journal on Applied Mathematics,2007,68(2):562-573. doi: 10.1137/06066641X [16] Zhang P, Wong S C, Shu C W. A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway[J].Journal of Computational Physics,2006,212(2):739-756. doi: 10.1016/j.jcp.2005.07.019 [17] Wong S C, Wong G C K. An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed-density relationship[J].Transportation Research Part B,2002,36(8):683-706. doi: 10.1016/S0191-2615(01)00023-6 [18] Karlsen K H, Risebro N H, Towers J D. Front tracking for scalar balance equations[J]. Journal of Hyperbolic Differential Equations,2004,1(1):115-148. doi: 10.1142/S0219891604000068 [19] Chen W, Wong S C, Shu C W. Efficient implementation of the shock-fitting algorithm for the Lighthill-Whitham-Richards traffic flow model[J].International Journal for Numerical Methods in Engineering,2007,74(4):554-600. [20] Jiang R, Hu M B, Jia B,et al. Enhancing highway capacity by homogenizing traffic flow[J].Transportmetrica,2008,4(1):51-61. doi: 10.1080/18128600808685676 [21] Zhang P, Wong S C, Xu Z. A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking[J].Computer Methods in Applied Mechanics Engineering,2008, 197(45/48):3816-3827. doi: 10.1016/j.cma.2008.03.003 [22] Shu C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[A]. In:Cockburn B,Johnson C,Shu C W, et al, Eds.Numerical Approximation of Nonlinear Hyperbolic Equations[C]. Vol 1697.Lecture Notes in Mathematics.Berlin,Heidelberg:Springe,1998, 325-432. [23] Zhang M, Shu C W, Wong G C K,et al. A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model[J].Journal of Computational Physics,2003, 191(2):639-659. doi: 10.1016/S0021-9991(03)00344-9
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