基于WENO-Z+重构的熵稳定格式求解LWR交通流模型
|
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.
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.
[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.