A Conserved HighOrder Traffic Flow Model With Discontinuous Flux and Its Numerical Simulation
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摘要: 在非均匀道路条件下,推广了各向异性守恒高阶交通流模型(CHO模型),获得流通量间断CHO模型,并基于其Riemann不变量性质,运用局部简化方法及δ映射算法,设计了求解流通量间断CHO模型的一阶Godunov、EO(Engquist-Osher)和LF(Lax-Friedrichs)等数值格式.通过数值模拟表明流通量间断CHO模型是合理有效的,它可以描述平衡态和非平衡态交通流,相对于流通量间断LWR(Lighthill-Whitham-Richards)模型,其能更好地刻画实际交通现象.
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关键词:
- 流通量间断CHO模型 /
- Riemann不变量 /
- 局部简化 /
- δ映射
Abstract: Under inhomogeneous road conditions, a conserved high-order (CHO) anisotropic traffic flow model was extended to obtain a CHO model with discontinuous fluxes. Based on the property of the Riemann invariant for the CHO model with discontinuous fluxes, the first-order Godunov, EO (Engquist-Osher) and LF (Lax-Friedrichs) numerical schemes for this model were designed with the local simplification method and the δ mapping algorithm. The numerical simulations show that, the CHO model with discontinuous fluxes is reasonable and effective. It can describe equilibrium and non-equilibrium traffic flows, and can better describe the actual traffic phenomena compared with the LWR (Lighthill-Whitham-Richards) model with a discontinuous flux. -
[1] LEBACQUE J P. The Godunov scheme and what it means for first order traffic flow models[C]// Proceedings of the 13th International Symposium on Transportation and Traffic Theory . Lyon, France, 1996: 647-677. [2] BALE D S, LEVEQUE R J. A Wave Propagation Method for Conservation Laws and Balance Laws With Spatially Varying Flux Functions [M]. Philadelphia: Society for Industrial and Applied Mathematics, 2002: 955-978. [3] LEVEQUE R J. Finite Volume Methods for Hyperbolic Problems [M]. Cambridge: Cambridge University Press, 2002. [4] LEVEQUE R J. Finite-volume methods for non-linear elasticity in heterogeneous media[J]. International Journal for Numerical Methods in Fluids,2002,40(1/2): 93-104. [5] LEVEQUE R J, YONG D H. Solitary waves in layered nonlinear media[J]. SIAM Journal on Applied Mathematics,2003,63(5): 1539-1560. [6] ZHANG P, LIU R X. 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. [7] ZHANG P, LIU R X. 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. [8] ZHANG P, LIU R X. Hyperbolic conservation laws with space-dependent fluxes II: general study of numerical fluxes[J]. Journal of Computational and Applied Mathematics,2005,176(1): 105-129. [9] XU Z L, ZHANG P, LIU R X. δ -mapping algorithm coupled with WENO reconstruction for nonlinear elasticity in heterogeneous media[J]. Applied Numerical Mathematics,2007,57(1): 103-116. [10] JIN W L, CHEN L, PUCKETT E G. Supply-demand diagrams and a new framework for analyzing the inhomogeneous Lighthill-Whitham-Richards model[C]//Transportation and Traffic Theory 2009: Golden Jubilee(ISTTT18) . USA, 2009: 603-635. [11] BRGER R, KARLSEN K H, TOWERS J D. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux[J]. Networks and Heterogeneous Media,2010,5(3): 461-485. [12] WANG G D. An Engquist-Osher type finite difference scheme with a discontinuous flux function in space[J]. Journal of Computational and Applied Mathematics,2011,235(17): 4966-4977. [13] CHEN J Z, SHI Z K, HU Y M. Numerical solutions of a multi-class traffic flow model on an inhomogeneous highway using a high-resolution relaxed scheme[J]. Journal of Zhejiang University: Science C (Computers & Electronics),2012,13(1): 29-36. [14] WIENS J K, STOCKIE J M, WILLIAMS J F. Riemann solver for a kinematic wave traffic model with discontinuous flux[J]. Journal of Computational Physics,2013,242: 1-23. [15] 张鹏, 王卓, 黄仕进. 交通流流体力学模型与非线性波[J]. 应用数学和力学, 2013,34(1): 85-97. (ZHANG Peng, WANG Zhuo, WONG S C. Fluid dynamics traffic flow models and their related non-linear waves[J]. Applied Mathematics and Mechanics,2013,34(1): 85-97. (in Chinese)) [16] QIAO D L, ZHANG P, LIN Z Y, et al. A Runge-Kutta discontinuous Galerkin scheme for hyperbolic conservation laws with discontinuous fluxes[J]. Applied Mathematics and Computation,2017,292(1): 309-319. [17] ZHANG P, LIU R X, DAI S Q. δ-mapping algorithm and its application in traffic flow problems with inhomogeneities[J]. Journal of Shanghai University (English Edition),2003,7(4): 315-317. [18] ZHANG P, LIU R X, WONG S C. High-resolution numerical approximation of traffic flow problems with variable lanes and free-flow velocities[J]. Physical Review E,2005,71(5): 056704. [19] BURGER R, GARCA A, KARLSEN K H, et al. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model[J]. Networks and Heterogeneous Media,2008,3(1): 1-41. [20] 张鹏, 吴冬艳, 黄仕进, 等. 交通流瓶颈效应的运动学描述[J]. 应用数学和力学, 2009,30(4): 399-408.(ZHANG Peng, WU Dongyan, WONG S C, et al. Kinetic description of bottleneck effects in traffic flow[J]. Applied Mathematics and Mechanics,2009,30(4): 399-408.(in Chinese)) [21] BURGER R, GARCA A, KARLSEN K, et al. A family of numerical schemes for kinematic flows with discontinuous flux[J]. Journal of Engineering Mathematics,2008,60(3/4): 387-425. [22] 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. [23] ZHANG P, WONG S, XU Z. A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking[J]. Computer Methods in Applied Mechanics and Engineering,2008,197(45/48): 3816-3827. [24] ZHANG P, WONG S, DAI S Q. A note on the weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model[J]. International Journal for Numerical Methods in Biomedical Engineering,2009,25(11): 1120-1126. [25] 张鹏, 乔殿梁, 李书峰. 流通量间断双曲守恒问题的推广WENO有限体格式[J]. 上海大学学报(自然科学版), 2009,15(6): 594-599. (ZHANG Peng, QIAO Dianliang, LI Shufeng. Extended finite volume WENO scheme for solving hyperbolic conservation laws with discontinuous fluxes[J]. Journal of Shanghai University (Natural Science),2009,15(6): 594-599. (in Chinese)) [26] BRGER R, MULET P, VILLADA L M. A diffusively corrected multiclass Lighthill-Whitham-Richards traffic model with anticipation lengths and reaction times[J]. Advances in Applied Mathematics & Mechanics,2013,5(5): 728-758. [27] ZHANG H M. A non-equilibrium traffic model devoid of gas-like behavior[J]. Transportation Research Part B: Methodological,2002,36(3): 275-290. [28] LEBACQUE J P, MAMMAR S, SALEM H H. Second order traffic flow modeling: the Riemann problem resolution using supply/demand based approach[C]//Proceedings of the Euro Working Group on Transportation. Poznan: Polish Academy of Sciences, 2005. [29] LEBACQUE J P, MAMMAR S, HAJ-SALEM H. The Aw-Rascle and Zhang’s model: vacuum problems, existence and regularity of the solutions of the Riemann problem[J]. Transportation Research Part B: Methodological,2007,41(7): 710-721. [30] LEBACQUE J P, MAMMAR S, SALEM H H. Generic second order traffic flow modelling[C]// Proceeding of the Seventeenth International Symposium on Transportation and Traffic Flow Theory . London, 2007: 755-766. [31] MAMMAR S, LEBACQUE J P, SALEM H H. Riemann problem resolution and Godunov scheme for the Aw-Rascle-Zhang model[J]. Transportation Science,2009,43(4): 531-545. [32] ZHANG P, WONG S C, DAI S Q. A conserved higher-order anisotropic traffic flow model: description of equilibrium and non-equilibrium flows[J]. Transportation Research Part B: Methodological,2009,43(5): 562-574. [33] ZHANG P, QIAO D L, DONG L Y, et al. A number of Riemann solvers for a conserved higher-order traffic flow model[C]// The Fourth International Joint Conference on Computational Sciences and Optimization . Kunming, 2011: 1049-1053. [34] QIAO D L, ZHANG P, WONG S C, et al. Discontinuous Galerkin finite element scheme for a conserved higher-order traffic flow model by exploring Riemann solvers[J]. Applied Mathematics & Computation,2014, 244(2): 567-576. [35] LIGHTHILL M, WHITHAM G. On kinematic waves Ⅱ: a theory of traffic flow on long crowded roads[J]. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences,1955,229(1178): 317-345. [36] RICHARDS P. Shock waves on the highway[J]. Operations Research,1956,4(1): 42-51. [37] WHITHAM G. Linear and Nonlinear Waves [M]. New York: Wiley New York, 1974. [38] SMOLLER J. Shock Waves and Reaction-Diffusion Equations [M]. New York: Springer, 1994. [39] TORO E. Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction [M]. Berlin: Springer Verlag, 1999. [40] KERNER B S, KONHAUSER P. Structure and parameters of clusters in traffic flow[J]. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics,1994,50(1): 54-83.
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