摘要
构造了交通流LWR模型方程相应的熵稳定格式.在数值模拟时,单元交界面的离散采用五阶WENO-Z+重构,时间方向的推进采用强稳定的三步三阶Runge-Kutta方法,从而得到了一种高精度、高分辨率以及数值稳定的熵稳定WENO-Z+格式.将新构造的熵稳定WENO-Z+数值算法应用于多个实际交通流问题的求解中,结果显示,该算法对激波有良好的捕捉效果,在间断区域没有非物理振荡,是模拟交通流LWR模型的理想方法.
In this paper,an entropy stable scheme for the LWR model in traffic flow is derived.By using higher order WENO-Z+ reconstruction at cell interfaces,a high resolution entropy stable WENO-Z+ scheme with numerical stability is obtained.The new constructed numerical method is applied to solve many practical traffic flow problems.The results show that entropy stable of WENO-Z+ scheme can capture shock waves well,and there is no unphysical oscillation in discontinuous areas.It is an ideal way to simulate traffic flow LWR model.
引文
[1] KINZER J P.Application of the Theory of Probability of Highway Traffic[D].New York:Polytechnic Institute of Brooklyn,1933.
[2] WONG G C K,WONG S C.A multi class traffic flow model an extension of LWR model with heterogeneous drivers[J].Transportation Res A,2002(36):827.
[3] LAX P D.Weak solutions of non-linear hyperbolic equations and their numerical computations[J].Comm Pure Appl Math,1954(7):159.
[4] CHEN J Z,HU Y M.A relaxation scheme for a multi-class Lighthill Whitham Richards traffic flow model[J].Journal of Zhejiang University:SCIENCE-A,2009,10(12):1835.
[5] GOTTLIEB S,SHU C W,TADMOR E.High order time discretizations with strong stability properties[J].SIAM Review,2001,43:89.
[6] LAX P D.Hyperbolic systems of conservation laws and the mathematical theory of shock waves[R].New York:Society for Industrial and Applied Mathematics,1973.
[7] TADMOR E.Numerical viscosity and the entropy conditions for conservative difference schemes[J].Math Comp,1984,43(168):369.
[8] ISMAIL F,ROE P L.Affordable,entropy-consistent Euler flux function:Entropy production at shocks[J].J Comput Phys,2009,228:5410.
[9] ISMAIL F.Towards a Reliable Prediction of Shocks in Hyperbolic Flow:Resolving Carbuncles with Entropy and Vorticity Control[D].Michigan:University of Michigan,2006.
[10] FJORDHOLM U,MISHRA S,TADMOR E.Energy preserving and energy stable schemes for the shallow water equations[C]//CUKER F,PINKUS A,TODD M J.Foundations of Computational Mathematics.London Mathematical Society Lecture Notes Series(363).London:Cambridge University Press,2009:93.
[11] CASTRO M,COSTA B,DON W S.High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws[J].J Comput Phys,2011,230(5):1766.
[12] WU X,ZHAO Y.A high-resolution hybrid scheme for hyperbolic conservation laws[J].Int J Numer,2015,78(3):162.
[13] ACKER F,BORGES R B D R,COSTA B.An improved WENO-Z scheme[J].J Comput Phys,2016,313:726.
[14] GANDE N R,RATHOD Y,RATHAM S.Third-order WENO-Z scheme with a new smoothness indicator[J].Numerical Method in Fluids,2017,85(2):171.
[15] 于玲.交通流流体动力学模型及数值模拟[D].广州:华南理工大学,2005.