Conservation Form of Helbing’s Fluid Dynamic Traffic Flow Model
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摘要: 得到了Helbing交通流流体力学模型的标准守恒形式,并证明了模型的双曲性,这对研究模型的解析性质和数值格式至关重要.基于给出的守恒形式,设计了高效求解模型方程的LDG(local discontinuous Galerkin)格式,并模拟了由不稳定平衡态到稳定的时停时走波的演化.数值模拟也表明,通过扩散系数校正确实使模型得到改进,避免了车辆碰撞和出现极端高密度.Abstract: A standard conservation form was derived,the hyperbolicity of Helbing's fluid dynamic traffic flow model was proved,which was essential for general analytical and numerical study of this model.On the basis of this conservation form,a local discontinuous Galerkin scheme is designed to solve the resulting model efficiently.The evolution of an unstable equilibrium traffic state leading to a stable stop-and-go traveling wave was simulated.This simulation also verifies that the model has been truly improved through the introduction of modified diffusion coefficients,thereby helping to protect vehicles from collisions and avoiding the appearance of extremely large density.
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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].Proc Roy Soc A, 1995, 229(1178): 317-345. [2] Richards P I. Shock waves on the highway[J].Operations Research, 1956, 4(1): 42-51. [3] Payne H J. Models of freeway traffic and control[C]Bekey A G.Mathematical Models of Public Systems.Simulation Council Proc, La Jolla, 1971, 1: 51-61. [4] Whitham G B. Linear and Nonlinear Waves[M].New York: John Wiley and Sons, 1974. [5] Kerner B S, Konhuser P. Structure and parameters of clusters in traffic flow[J].Phys Rev E, 1994, 50(1): 54-83. doi: 10.1103/PhysRevE.50.54 [6] Siebel F, Mauser W. On the fundamental diagram of traffic flow[J].SIAM J Appl Math, 2006, 66(4): 1150-1162. doi: 10.1137/050627113 [7] Zhang P, Wong S C, Dai S Q. Characteristic parameters of a wide cluster in a higher-order traffic flow model[J].Chin Phys Lett, 2006, 23(2): 516-519. doi: 10.1088/0256-307X/23/2/067 [8] Zhang P, Wong S C. Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models[J].Phys Rev E, 2006, 74(2): 026109. doi: 10.1103/PhysRevE.74.026109 [9] Xu R Y, Zhang P, Dai S Q, Wong S C. Admissibility of a wide cluster solution in anisotropic higher-order traffic flow models.[J].SIAM J Appl Math, 2007, 68(2): 562-573. doi: 10.1137/06066641X [10] Zhang P, Wong S C, Dai S Q. A conserved higher-order anisotropic traffic flow model: description of equilibrium and non-equilibrium flows[J].Transpn Res Part B, 2009, 43(5): 562-574. doi: 10.1016/j.trb.2008.10.001 [11] Tang T Q, Huang H J, Shang H Y. A new macro model for traffic flow with the consideration of the driver’s forecast effect[J].Physics Letters A, 2010, 374(15/16): 1668-1672. doi: 10.1016/j.physleta.2010.02.001 [12] Prigogine I, Herman R. Kinetic Theory of Vehicular Traffic[M].New York: American Elsevier Publishing Co,1971. [13] Paveri-Fontana S L. On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis[J].Transpn Res, 1975, 9(4): 225-235. doi: 10.1016/0041-1647(75)90063-5 [14] Phillips W. Kinetic Model for Traffic Flow[M].Springfield, VA: National Technical Information Service, 1977. [15] Helbing D. A fluid-dynamic model for the movement of pedestrians[J].Complex Systems, 1992, 6(5): 391-415. [16] Helbing D, Hennecke A, Shvetsov V, Treiber M. Master: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model[J]. Transpn Res Part B, 2001, 35(2): 183-211. doi: 10.1016/S0191-2615(99)00047-8 [17] Hoogendoorn S P, Bovy P H L. Continuum modeling of multiclass traffic flow[J]. Transpn Res Part B, 2000, 34(2): 123-146. doi: 10.1016/S0191-2615(99)00017-X [18] Hoogendoorn S P, Bovy P H L. Generic gas-kinetic traffic systems modeling with applications to vehicular traffic flow[J]. Transpn Res Part B, 2001, 35(4): 317-336. doi: 10.1016/S0191-2615(99)00053-3 [19] Ngoduy D. Derivation of continuum traffic model for weaving sections on freeways[J]. Transportmetrica, 2006, 2(3): 199-222. doi: 10.1080/18128600608685662 [20] Ngoduy D. Application of gas-kinetic theory to modeling mixed traffic of manual and ACC vehicles[J]. Transportmetrica, doi: 10.1080/18128600903578843. [21] Helbing D. Improved fluid-dynamic model for vehicular traffic[J].Phys Rev E, 1995, 51(4): 3164-3169. [22] Liu T P. Hyperbolic and viscous conservation laws[C]CBMS-NSF Regional Conference Series in Applied Mathematics 72, SIAM, Philadelphia, PA, 2000. [23] Cockburn B, Lin S Y, Shu C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws Ⅲ: one dimensional systems[J].J Comput Phys, 1989, 84(1): 90-113. doi: 10.1016/0021-9991(89)90183-6 [24] Cockburn B, Shu C W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems[J].SIAM J Numer Anal, 1998, 35(6): 2440-2463. doi: 10.1137/S0036142997316712 [25] Bassi F, Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations[J].J Comput Phys, 1997, 131(2): 267-279. doi: 10.1006/jcph.1996.5572
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