交通流的介观与微观模型及其应用
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摘要
本文从介观和微观角度研究交通流的建模和模拟。面对我国交通平面、混合与低速的实际特征以及交通规划、管理实践中的信息化与智能化的趋势,基于Nagel-Schreckenberg(NaSch)元胞自动机模型,建立了适用于汽车-摩托车混合交通的改进型模型,并针对一类较为实际的双路径交通情境研究了实时信息反馈对交通系统的影响;为了克服传统的介观模型中包含复杂的积分微分项所导致的困难,建立了道路交通流的格子玻尔兹曼模型,并将其推广到城市网络交通流的建模研究中。相关的模拟表明,所建立的模型可以再现交通流的复杂非线性动力学特性。
全文主要内容如下:
一.汽车与摩托车混合交通建模研究
基于划分“虚拟子车道”的思想,推广了描述单车道汽车流的NaSch模型,首次尝试建立了汽车-摩托车混合交通流模型。通过在周期性边界条件下进行的数值模拟,详细考察了这种混合交通流的流量-密度关系以及摩托车的“换道”行为。模拟发现,在摩托车“换道”行为的影响下,最大汽车流量明显降低,而最大总体车流量则随着摩托车密度的增加而先增后减,且模型中总体车流由自由流向拥挤交通的相变是平滑的。摩托车“换道”率随汽车密度演化的趋势相当复杂,但最终将随着汽车密度的增加而趋于零。所发现的另一个令人感兴趣的事实是:随着摩托车密度增加,摩托车“换道”率先增后减,与前人在多车道纯汽车流中发现的现象十分相似。当摩托车密度较小时,“换道”行为几乎无助于提高摩托车流量。但是,当摩托车密度足够大时,“换道”行为可显著提高摩托车流量,而且这时随汽车密度增加,摩托车流量逐渐降低并趋于与摩托车NaSch模型一致。数值模拟结果表明,除非摩托车密度甚大而汽车密度甚小,实施摩托车与汽车分道行驶确有必要。
二.双路径交通流决策动力学研究
针对一类包含长短不同路径的双路径交通情境,基于NaSch元胞自动机模型,讨论了各种信息反馈策略对交通系统的影响。分析了现有策略的不足之处,提出了一类预测策略与无预测策略联合使用的新途径。在适当的开放边界条件下进行的数值模拟显示,与无信息反馈相比,信息反馈策略可在一定程度上提高交通效率,但各种信息反馈策略的效果并不相同。一般而言,双路径策略优于单路径策略,有预测策略优于无预测策略。除了一些有预测策略以外,其它策略都会引起平均密度和平均速度的振荡。此外,还考察了各种信息反馈策略对双路径系统中长路径的影响。模拟结果还表明,各种信息反馈策略都或多或少地对两条路径中长路径上的交通产生影响,不过,有预测策略产生的影响通常小于无预测策略。但在长路径上的交通负载过重的情况下,即使是有预测策略在提高交通效率等方面也无明显效果。本项研究证实:控制交通的正确诱导策略极其重要。
三.道路交通流的格子玻尔兹曼模型建模研究
由于存在复杂的积分微分项,直接应用传统的介观交通流模型通常非常困难。因此,本文基于类Bhatnagar-Gross-Krook(BGK)近似与时间和相空间的离散化,建立了道路交通流的格子玻尔兹曼模型(lattice Boltzmann model,LBM)。该模型属于离散模型,具有形式简洁、参数物理意义明确的特点。因此,利用该模型可以方便地实现交通流模拟。随后,文中利用Taylor展开与Chapman-Enskog展开考察了模型的宏观动力学特性。为检验模型的有效性,在周期性边界条件下进行了数值模拟。结果显示:模型能够合理地再现基本图,且能够捕捉一些基本的非线性物理现象,如亚稳态和时停时走交通等。结果表明,格子玻尔兹曼模型是一种行之有效的交通模型。
四.基于格子玻尔兹曼模型的城市网络交通流建模研究
揭示道路交通网络的动力学特性具有重要意义,目前在介观层次上研究交通网络的工作尚不多见。基于将Biham-Middleton-Levine(BML)元胞自动机模型与路段格子玻尔兹曼模型相耦合的思想,我们建立了一个适用于城市网络交通流的格子玻尔兹曼模型。该模型通过将具有红绿灯的交叉口视为随时间演变的边界条件来实现对城市网络交通流的介观描述。通过数值模拟,详细考察了平均速度随时间的演化行为,得到了与Chowdbury-Schadschneider(ChSch)元胞自动机模型相符的结果,而且由于我们的离散模型具有统计噪声较小的特征,因而具有较高的计算效率。研究表明,只要引用合理的假设和技巧,介观模型在交通流研究中大有用武之地。
This thesis is concerned with the modeling and simulation of traffic flows from mesoscopic and microscopic viewpoints. In view of the realistic characteristics of plane, low-velocity and mixed transportation systems in China and the trend of widely applying real-time information in traffic planning and management, based on the Nagel-Schreckenberg (NaSch) cellular automaton (CA) model, an improved CA model is proposed for mixed traffic flow with motorcycles, and the decision dynamics in a realistic two-route scenario is investigated. Then in order to overcome the difficulties in applying the traditional mesoscopic models with the appearance of integro-differential terms in the models, a lattice Boltzmann model for road traffic flows is established and then extended to modeling and simulating urban network traffic flows. The related simulations show that the presented models can be employed to reproduce the complicated nonlinear dynamic characteristics in traffic flows.
The main contents of the dissertation are as follows.
I. Cellular automaton model for the mixed traffic flow with motorcycles
A single-lane cellular automaton model is firstly proposed to simulate the traffic flow of cars mixed with motorcycles by dividing a single lane into three virtual sub-lanes and extending the NaSch model for single-lane car flows. Through performing numerical simulations under the periodic boundary conditions, some flow-density relations and the "lane-changing"behavior of motorcycles are investigated in detail. It is found that the maximum car flow remarkably decreases due to the "lane-changing "behavior of motorcycles, while the maximum total flow increases first and then decreases with increasing motorcycle density. Moreover, the phase transition of the total flow from the free flow to the congested flow is smooth in this model. The "lane-changing" rate of motorcycles will finally decrease to zero with the increase of the car density. But its evolutionary trend is considerably complex. Another interesting fact found in the simulation is that, with the increase of the motorcycle density, the "lane-changing" rate increases first and decreases later. This phenomenon is very similar to the findings in previous work on multi-lane pure car flows. The "lane-changing" is almost of no use in increasing the flow of motorcycles as the motorcycle density is small. But it distinctly causes the increase in the flow of motorcycles as the motorcycle density is sufficiently high, and in this density regime, the flow of motorcycles gradually decreases to the one given with the NaSch model for motorcycles with the increase of the car density. The simulation results indicate that it is necessary to set a barrier or a partition lane for separating the motorcycle flow from the car flow except for the situation of higher motocycle density and lower car density.
II. Decision dynamics in a realistic two-route scenario
The optimal information feedback is of great importance in making full use of the existing transportation resources and improving the performance of traffic systems. Thus, several information feedback strategies were proposed in literature and their effects on the traffic systems were also investigated. However, there is still a paradox remaining in some research reports that no information feedback strategy seems to be the best ones. For examining this paradox, a further study is conducted with a realistic two-route traffic scenario containing a longer route and a shorter route and appropriate open boundary conditions. The effects of several previous strategies on traffic systems are reconsidered in detail. Meanwhile, through analyzing the shortcomings of the existing strategies, a new approach of combing a prediction strategy with various prediction-free strategies is proposed in this work. By conducting numerical simulations, it is found that all the strategies can more or less improve the efficiency compared with the situation with no information feedback. However, various information feedback strategies might have different consequences. Generally speaking, two-route strategies are superior to single-route ones, and strategies based on prediction are better than prediction-free ones. Except for several strategies with prediction, most of the strategies will cause the oscillations of average density and velocity. In addition, the influence of various strategies on the traffic on the longer route is studied, which proves to exist in general, but less remarkable for the strategies with prediction. Especially, the appropriate strategy with prediction is helpful for improving the traffic efficiency and stabilizing the traffic flow. Nevertheless, further simulations show that the information feedback is not very helpful for improving the efficiency if the traffic load on the longer route is too heavy. This study shows that it is essential to correctly choose information feedback strategies.
III. Lattice Boltzmann model for road traffic flow
Traditional mesoscopic models for traffic flows are usually difficult to be directly employed because of the appearance of integro-differential terms in the models. Thus, a more applicable lattice Boltzmann model (LBM) for road traffic flows is constructed on the basis of the Bhatnagar-Gross-Krook (BGK)-like approximation for the Boltzmann equation and its discretization in time and phase-space. The so-obtained model is a simpler discrete version of the gas kinetics model with the physically meaningful distinct parameters, which can be easily used to investigate numerically the behavior of traffic flows. In consequence, the macroscopic dynamics of the model is derived through the Taylor expansion and Chapman-Enskog expansion. For validating the model, numerical simulations are conducted under the periodic boundary conditions. It is found that the presented model could reasonably reproduce the fundamental diagram. Moreover, certain interesting physical phenomena can be captured, such as the metastability and stop-and-go phenomena, etc. The results imply that the presented model is one of the effective and efficient traffic models.
IV. Modeling of urban network traffic flows with lattice Boltzmann model
It is of great importance to uncover the characteristics of traffic networks. However, there have appeared few researches concerning kinetics models for traffic networks. Thus, an LBM for road traffic networks is proposed by incorporating the ideas of the Biham-Middleton-Levine (BML) CA model into the LBM for road traffic. In the present model, situations at intersections with the traffic signals are treated as a kind of boundary conditions varying with time. Thus, the network traffic flow could be described in the mesoscopic level. By performing numerical simulations under the periodic boundary conditions, the evolution of average velocity is investigated in detail. The numerical results agree quite well with those given by the Chowdhury-Schadschneider (ChSch) CA model. Furthermore, the statistical noise is reduced in this discrete kinetics model, and thus the present model has considerably higher computational efficiency. This study shows that the meso-scopic traffic model could find its extensive applications, provided that reasonable assumptions and techniques are introduced.
全文主要内容如下:
一.汽车与摩托车混合交通建模研究
基于划分“虚拟子车道”的思想,推广了描述单车道汽车流的NaSch模型,首次尝试建立了汽车-摩托车混合交通流模型。通过在周期性边界条件下进行的数值模拟,详细考察了这种混合交通流的流量-密度关系以及摩托车的“换道”行为。模拟发现,在摩托车“换道”行为的影响下,最大汽车流量明显降低,而最大总体车流量则随着摩托车密度的增加而先增后减,且模型中总体车流由自由流向拥挤交通的相变是平滑的。摩托车“换道”率随汽车密度演化的趋势相当复杂,但最终将随着汽车密度的增加而趋于零。所发现的另一个令人感兴趣的事实是:随着摩托车密度增加,摩托车“换道”率先增后减,与前人在多车道纯汽车流中发现的现象十分相似。当摩托车密度较小时,“换道”行为几乎无助于提高摩托车流量。但是,当摩托车密度足够大时,“换道”行为可显著提高摩托车流量,而且这时随汽车密度增加,摩托车流量逐渐降低并趋于与摩托车NaSch模型一致。数值模拟结果表明,除非摩托车密度甚大而汽车密度甚小,实施摩托车与汽车分道行驶确有必要。
二.双路径交通流决策动力学研究
针对一类包含长短不同路径的双路径交通情境,基于NaSch元胞自动机模型,讨论了各种信息反馈策略对交通系统的影响。分析了现有策略的不足之处,提出了一类预测策略与无预测策略联合使用的新途径。在适当的开放边界条件下进行的数值模拟显示,与无信息反馈相比,信息反馈策略可在一定程度上提高交通效率,但各种信息反馈策略的效果并不相同。一般而言,双路径策略优于单路径策略,有预测策略优于无预测策略。除了一些有预测策略以外,其它策略都会引起平均密度和平均速度的振荡。此外,还考察了各种信息反馈策略对双路径系统中长路径的影响。模拟结果还表明,各种信息反馈策略都或多或少地对两条路径中长路径上的交通产生影响,不过,有预测策略产生的影响通常小于无预测策略。但在长路径上的交通负载过重的情况下,即使是有预测策略在提高交通效率等方面也无明显效果。本项研究证实:控制交通的正确诱导策略极其重要。
三.道路交通流的格子玻尔兹曼模型建模研究
由于存在复杂的积分微分项,直接应用传统的介观交通流模型通常非常困难。因此,本文基于类Bhatnagar-Gross-Krook(BGK)近似与时间和相空间的离散化,建立了道路交通流的格子玻尔兹曼模型(lattice Boltzmann model,LBM)。该模型属于离散模型,具有形式简洁、参数物理意义明确的特点。因此,利用该模型可以方便地实现交通流模拟。随后,文中利用Taylor展开与Chapman-Enskog展开考察了模型的宏观动力学特性。为检验模型的有效性,在周期性边界条件下进行了数值模拟。结果显示:模型能够合理地再现基本图,且能够捕捉一些基本的非线性物理现象,如亚稳态和时停时走交通等。结果表明,格子玻尔兹曼模型是一种行之有效的交通模型。
四.基于格子玻尔兹曼模型的城市网络交通流建模研究
揭示道路交通网络的动力学特性具有重要意义,目前在介观层次上研究交通网络的工作尚不多见。基于将Biham-Middleton-Levine(BML)元胞自动机模型与路段格子玻尔兹曼模型相耦合的思想,我们建立了一个适用于城市网络交通流的格子玻尔兹曼模型。该模型通过将具有红绿灯的交叉口视为随时间演变的边界条件来实现对城市网络交通流的介观描述。通过数值模拟,详细考察了平均速度随时间的演化行为,得到了与Chowdbury-Schadschneider(ChSch)元胞自动机模型相符的结果,而且由于我们的离散模型具有统计噪声较小的特征,因而具有较高的计算效率。研究表明,只要引用合理的假设和技巧,介观模型在交通流研究中大有用武之地。
This thesis is concerned with the modeling and simulation of traffic flows from mesoscopic and microscopic viewpoints. In view of the realistic characteristics of plane, low-velocity and mixed transportation systems in China and the trend of widely applying real-time information in traffic planning and management, based on the Nagel-Schreckenberg (NaSch) cellular automaton (CA) model, an improved CA model is proposed for mixed traffic flow with motorcycles, and the decision dynamics in a realistic two-route scenario is investigated. Then in order to overcome the difficulties in applying the traditional mesoscopic models with the appearance of integro-differential terms in the models, a lattice Boltzmann model for road traffic flows is established and then extended to modeling and simulating urban network traffic flows. The related simulations show that the presented models can be employed to reproduce the complicated nonlinear dynamic characteristics in traffic flows.
The main contents of the dissertation are as follows.
I. Cellular automaton model for the mixed traffic flow with motorcycles
A single-lane cellular automaton model is firstly proposed to simulate the traffic flow of cars mixed with motorcycles by dividing a single lane into three virtual sub-lanes and extending the NaSch model for single-lane car flows. Through performing numerical simulations under the periodic boundary conditions, some flow-density relations and the "lane-changing"behavior of motorcycles are investigated in detail. It is found that the maximum car flow remarkably decreases due to the "lane-changing "behavior of motorcycles, while the maximum total flow increases first and then decreases with increasing motorcycle density. Moreover, the phase transition of the total flow from the free flow to the congested flow is smooth in this model. The "lane-changing" rate of motorcycles will finally decrease to zero with the increase of the car density. But its evolutionary trend is considerably complex. Another interesting fact found in the simulation is that, with the increase of the motorcycle density, the "lane-changing" rate increases first and decreases later. This phenomenon is very similar to the findings in previous work on multi-lane pure car flows. The "lane-changing" is almost of no use in increasing the flow of motorcycles as the motorcycle density is small. But it distinctly causes the increase in the flow of motorcycles as the motorcycle density is sufficiently high, and in this density regime, the flow of motorcycles gradually decreases to the one given with the NaSch model for motorcycles with the increase of the car density. The simulation results indicate that it is necessary to set a barrier or a partition lane for separating the motorcycle flow from the car flow except for the situation of higher motocycle density and lower car density.
II. Decision dynamics in a realistic two-route scenario
The optimal information feedback is of great importance in making full use of the existing transportation resources and improving the performance of traffic systems. Thus, several information feedback strategies were proposed in literature and their effects on the traffic systems were also investigated. However, there is still a paradox remaining in some research reports that no information feedback strategy seems to be the best ones. For examining this paradox, a further study is conducted with a realistic two-route traffic scenario containing a longer route and a shorter route and appropriate open boundary conditions. The effects of several previous strategies on traffic systems are reconsidered in detail. Meanwhile, through analyzing the shortcomings of the existing strategies, a new approach of combing a prediction strategy with various prediction-free strategies is proposed in this work. By conducting numerical simulations, it is found that all the strategies can more or less improve the efficiency compared with the situation with no information feedback. However, various information feedback strategies might have different consequences. Generally speaking, two-route strategies are superior to single-route ones, and strategies based on prediction are better than prediction-free ones. Except for several strategies with prediction, most of the strategies will cause the oscillations of average density and velocity. In addition, the influence of various strategies on the traffic on the longer route is studied, which proves to exist in general, but less remarkable for the strategies with prediction. Especially, the appropriate strategy with prediction is helpful for improving the traffic efficiency and stabilizing the traffic flow. Nevertheless, further simulations show that the information feedback is not very helpful for improving the efficiency if the traffic load on the longer route is too heavy. This study shows that it is essential to correctly choose information feedback strategies.
III. Lattice Boltzmann model for road traffic flow
Traditional mesoscopic models for traffic flows are usually difficult to be directly employed because of the appearance of integro-differential terms in the models. Thus, a more applicable lattice Boltzmann model (LBM) for road traffic flows is constructed on the basis of the Bhatnagar-Gross-Krook (BGK)-like approximation for the Boltzmann equation and its discretization in time and phase-space. The so-obtained model is a simpler discrete version of the gas kinetics model with the physically meaningful distinct parameters, which can be easily used to investigate numerically the behavior of traffic flows. In consequence, the macroscopic dynamics of the model is derived through the Taylor expansion and Chapman-Enskog expansion. For validating the model, numerical simulations are conducted under the periodic boundary conditions. It is found that the presented model could reasonably reproduce the fundamental diagram. Moreover, certain interesting physical phenomena can be captured, such as the metastability and stop-and-go phenomena, etc. The results imply that the presented model is one of the effective and efficient traffic models.
IV. Modeling of urban network traffic flows with lattice Boltzmann model
It is of great importance to uncover the characteristics of traffic networks. However, there have appeared few researches concerning kinetics models for traffic networks. Thus, an LBM for road traffic networks is proposed by incorporating the ideas of the Biham-Middleton-Levine (BML) CA model into the LBM for road traffic. In the present model, situations at intersections with the traffic signals are treated as a kind of boundary conditions varying with time. Thus, the network traffic flow could be described in the mesoscopic level. By performing numerical simulations under the periodic boundary conditions, the evolution of average velocity is investigated in detail. The numerical results agree quite well with those given by the Chowdhury-Schadschneider (ChSch) CA model. Furthermore, the statistical noise is reduced in this discrete kinetics model, and thus the present model has considerably higher computational efficiency. This study shows that the meso-scopic traffic model could find its extensive applications, provided that reasonable assumptions and techniques are introduced.
引文
[戴世强等1997]戴世强,冯苏苇,顾国庆.1997.交通流动力学:它的内容、方法和意义自然杂志19:196-201.
[戴世强＆薛郁2004]戴世强,薛郁.2004.交通流的建模与仿真,交通流与颗粒流(唐孝威等主编).浙江大学出版社.
[冯蔚东等2003]冯蔚东,陈剑,贺国光,刘豹.2003.交通流中的分形研究.高技术通讯pp.59-65.
[付传技等2006]付传技,汪秉宏,殷传洋,高坤.2006.利用智能决策的双通道交通流.物理学报.55:4032-4038.
[甘明星等2004]甘明星,张小军,刘祖源.2004.交通流的格子Boltzmann模型.武汉理工大学学报(交通科学与工程版)28:453-455.
[甘明星2004]甘明星.2004.基于Lattice Boltzmann方法的交通流模型研究.武汉理工大学硕士学位论文.
[黄海军2004]黄海军.2004.交通流理论的创立.流体中文网论坛-交通流论坛
[黄乒花等2002]黄乒花,孔令江,刘慕仁.2002.用格子Boltzmann方法研究交通流BML模型.广西师范大学学报(自然科学版)20:6-10.
[黄乒花2001]黄乒花.2001.用元胞自动机和格子玻尔兹曼方法研究交通流的有关问题.广西师范大学硕士学位论文.
[任福田等2003]任福田,刘小明,荣建.2003.交通工程学.人民交通出版社.
[宋涛2006]宋涛.2006.城市地面-高架路交通复杂网络的超越图模型及动力学模拟.上海大学硕士学位论文.
[SCCTPI2005]2005.上海城市综合交通发展报告.技术报告,上海市城市综合交通规划研究所.
[汪秉宏等1998a]汪秉宏,邝乐琪,许伯铭.1998a.高速公路交通流元胞自动机模型的一种统计平均解耦处理.物理学报47:906-915.
[张鹏等2005]张鹏,戴世强,刘儒勋.2005.多等级交通流LWR模型中的非线性波描述与WENO数值逼近.应用数学和力学26:637-644.
[Aw and Rascle2000]Aw,A.AND RASCLE,M.2000.Resurrection of "second order" models of traffic flow.SIAM J.Appl.Math.60:916-938.
[Bando et al.1995a]BANDO,M.,HASEBE,K.,NAKANISHI,K.,NAKAYAMA,A.,SHIBATA,A.,AND SUGIYAMA,Y.1995a.Phenomenological study of dynamical model of traffic flow.J.Phys.I France 5:1389-1399.
[Bando et al.1995b]BANDO,M.,HASEBE,K.,NAKAYAMA,A.,SHIBATA,A.,AND SUGIYAMANO,V.1995b.Dynamical model of traffic congestion and numerical simulation.Phys.Rev.E 51:1035-1042.
[Barlovic et al.1998]BARLOVIC,R.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.1998.Metastable states in cellular automata for traffic flow.Eur.Phy.J.B 5:793-800.
[Bellomo et al.2002]BELLOMO,N.,DELITALA,M.,AND COSCIA,V.2002.On the mathematical theory of vehicular traffic flow I:Fluid dynamic and kinetic modelling.Math.Models Meth.Appl.Sci.12:1801-1843.
[Benjamin et al.1996]BENJAMIN,S.C.,JOHNSON,N.F.,AND HUI,P.M.1996.Cellular automata models of traffic flow along a highway containing a junction.J.Phys.A:Math.Gen.29:3119-3127.
[Benzi et al.1992]BENZI,R.,SUCCI,S.,AND VERGASSOLA,M.1992.The lattice Boltzmann equation:Theory and applications.Rhys.Rep 222:145-197.
[Bhatnagar et al.1954]BHATNAGAR,P.,GROSS,E.,AND KROOK,M.1954.A model for collision processs in gases.I.Small amplitude processess in charged and neutral one-component system.Phys.Rev 94:511-525.
[Biham et al.1992]BIHAM,O.,MIDDLETON,A.,AND LEVINE,D.1992.Selforgranization and a dynamical transition in traffic-flow models.Phys.Rev.A 46:R6124-R6127.
[Buchanan2008]BUCHANAN,M.2008.Physics is dead,long live physics! Nature physics 4:159.
[Chandler et al.1958]CHANDLER,R.,HERMAN,R.,AND MONTROLL,E.1958.Traffic dynamics:Studies in car following.Oper.Res.6:165-184.
[Chen et al.1992]CHEN,H.-D.,CHEN,S.-V.,AND MATTHAEUS,W.H.1992.Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method.Phys.Rev.A 45:R5339-R5342.
[Chen et al.2004]CHEN,W.,HUANG,D.,HUANG,W.,AND HWANG,W.2004.Traffic flow on a 3-lane highway.Int.J.Mod.Phys.B 18:4161-4171.
[Cheybani et al.2000a]CHEYBANI,S.,KERTESZ,J.,AND SCHRECKENBERG,M.2000a.Nondeterministic Nagel-Schreckenberg traffic model with open boundary conditions.Phys.Rev.E 63:016108(9).
[Cheybani et al.2000b]CHEYBANI,S.,KERTESZ,J.,AND SCHRECKENBERG,M.2000b.Stochastic boundary conditions in the deterministic Nagel-Schreckenberg traffic model.Phys.Rev.E 63:016107.
[Cho and Lo2002]CHO,H.-J.AND LO,S.-C.2002.Modeling self-consistent multi-class dynamic traffic flow.Physica A 312:342-362.
[Chopard et al.1996]CHOPARD,B.,LUTHI,P.,AND QUELOZ,P.1996.Cellular automata model of car traffic in a two-dimensional street network.J.Phys.A:Math.and Gen.29:2325-2336.
[Chowdhury et al.2000]CHOWDHURY,D.,SANTEN,L.,AND SCHADSCHNEIDER,A.2000.Statistical physics of vehicular traffic and some realted systems.Phys.Rep 329:199-329.
[Chowdhury and Schadschneider1999]CHOWDHURY,D.AND SCHADSCHNEIDER,A.1999.Self-organization of traffic jams in cities:Effects of stochastic dynamics and signal periods.Phys.Rev.E 59:R1311-R1314.
[Chowdhury et al.1997]CHOWDHURY,D.,WOLF,D.,AND SCHRECKENBERG,M.1997.Particle hopping models for two-lane traffic with two kinds of vehicles:Effects of lane-changing rules.Physica A 235:417-439.
[Chung et al.1995]CHUNG,K.-H.,HUI,P.-M.,AND Gu,G.-Q.1995.Twodimensional traffic flow problems with faulty traffic lights.Phys.Rev.E 51:772-774.
[Coclite et al.2005]COCLITE,G.,GARAVELLO,M.,AND PICCOLI,B.2005.Traffic flow on a road network.SIAM J.Math.Anal.36:1862-1886.
[Coscia et al.2007]COSCIA,V.,DELITALA,M.,AND FRASCA,P.2007.On the mathematical theory of vehicular traffic flow Ⅱ:Discrete velocity kinetic models.Int.J.Non-Linear Mech.42:411-421.
[Daganzo1995]DAGANZO,C.1995.Requiem for second-order fluid approximations of traffic flow.Transpn.Res.B 29B:277-286.
[Daganzo1994]DAGANZO,C.F.1994.The cell transmission model:A dynamic representation of highway traffic consistent with the hydrodynamic theory.Transpn.Res.B 28:269-287.
[Delitala2003]DELITALA,M.2003.Nonlinear models of vehicular traffic flownew frameworks of the mathematical kinetic theory.Comptes Rendus Mecanique 331:817-822.
[Ebersbach and Schneider2004]EBERSBACH,A.AND SCHNEIDER,J.2004.Twolane traffic with places of obstruction to traffic.Int.J.Mod.Phys.C 15:535-544.
[Ebersbach et al.2000]EBERSBACH,A.,SCHNEIDER,J.,MORGENSTERN,I.,AND HAMMWOHNER,R.2000.The influence of trucks on traffic flow-an investigation on the Nagel-Schreckenberg-model.Int.J.Mod.Phys.C 11:837-842.
[Ez-Zahraouya et al.2004]EZ-ZAHRAOUYA,H.,JETTO,K.,AND BENYOUSSEF,A.2004.The effect of mixture lengths of vehicles on the traffic flow behaviour in one-dimensional cellular automaton.Eur.Phys.J.B 40:111-117.
[Feng et al.1997]FENG,S.-W.,Gu,G.-Q.,AND DAI,S.-Q.1997.Effects of traffic lights on CA traffic model.Communications in Nonlinear Science and Numerical Simulation 2:70-74.
[Frisch et al.1986]FRISCH,U.,HASSLACHER,B.,AND POMEAU,Y.1986.Lattice-gas automata for the Navier-Stokes equation.Phys.Rev.Lett.56:1505-1508.
[Fukui and Ishibashi1996]FUKUI,M.AND ISHIBASHI,Y.1996.Traffic flow in 1D cellular automaton model including cars moving with high speed.J.Phys.Soc.Jpn 65:1868-1870.
[Garavello and Piccoli2006]GARAVELLO,M.AND PICCOLO,B.2006.Traffic Flow on Networks,volume 1 of AIMS Series on Applied Math.American Institute of Mathematical Sciences.
[Gartner et al.1999]N.Gartner,C.Messer,and A.Rathi(eds.) 1999.Revised monograph on traffic flow theory.The federal highway administration(FHWA).
[Gazis2002]GAZIS,D.2002.The origins of traffic theory.Oper.Res.50:69-77.
[Ge et al.2004]GE,H.-X.,DAI,S.-Q.,DONG,L.-Y.,AND XUE,Y.2004.Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application.Phys.Rev.E 70:066134.
[Gu et al.1995]Gu,G.-Q.,CHUNG,K.-H.,AND HUI,P.-M.1995.Twodimensional traffic flow problems in inhomogeneous lattices.Physica A 217:339-347.
[He and Luo1997a]HE,X.-Y.AND LUO,L.-S.1997a.A priori derivation of the lattice Boltzmann equation.Phys.Rev.E 55:6333(4).
[He and Luo1997b]HE,X.-Y.AND LUO,L.-S.1997b.Theory of the lattice Boltzmann method:from the Boltzmann equation to the lattice Boltzmann equation.Phys.Rev.E 56:6811-6817.
[Helbing1995]HELBING,D.1995.Improved fluid-dynamic model for vehicular traffic.Phys.Rev.E 51:3164-3169.
[Helbing1996]HELBING,D.1996.Gas-kinetic derivation of Navier-Stokes-like traffic equations.Phys.Rev.E 53:2366-2381.
[Helbing1997a]HELBING,D.1997a.Empirical traffic data and their implications for traffic modeling.Phys.Rev.E 55:25(4).
[Helbing1997b]HELBING,D.1997b.Modeling multi-lane traffic flow with queuing effects.Physica A 242:175-194.
[Helbing1998]HELBING,D.1998.From microscopic to macroscopic traffic models.arXiv:cond-mat/9806171v1.
[Helbing2001]HELBING,D.2001.Traffic and related self-driven many-particles systems.Rev.Mod.Phys 73:1067-1141.
[Helbing et al.2001]HELBING,D.,HENNECKE,A.,SHVETSOV,V.,AND TREIBER,M.2001.MASTER:macroscopic traffic simulation based on a gaskinetic,non-local traffic model.Transpn.Res.B 35:183-211.
[Helbing and Tilch1998]HELBING,D.AND TILCH,B.1998.Generalized force model of traffic dynamics.Phys.Rev.E 58:133-138.
[Holden and Risebro1995]HOLDEN,H.AND RISEBRO,N.H.1995.A mathematical model of traffic flow on a network of unidirectional roads.SIAM J.Math.Anal.26:999-1017.
[Hoogendoorn and Bovy2000]HOOGENDOORN,S.AND BOVY,P.2000.Continuum modeling of multiclass traffic flow.Transpn.Res.B 34:123-146.
[Hoogendoorn and Bovy2001]HOOGENDOORN,S.AND BOVY,P.2001.Generic gas-kinetic traffic systems modeling with applications to vehicular traffic flow.Transpn.Res.B 35:317-336.
[Hu et al.2003]Hu,T.-P.,AHMAD,F.,AND NGUYEN,X.-D.2003.A comparison study on motorcycle traffic development in some Asian countries-case of Taiwan,Malaysia and Vietnam.Technical report,Institute of Civil Engineering National Taiwan University.
[Huang and Huang2006]HUANG,D.-W.AND HUANG,W.-N.2006.Biham-Middleton-Levine model with four-directional traffic.Physica A 370:747-755.
[Jia et al.2004]JIA,B.,JIANG,R.,AND WU,Q.-S.2004.A realistic two-lane cellular automaton model for traffic flow.Int.J.Mod.Phys.C 15:381-392.
[Jiang and Wu2003]JIANG,R.AND WU,Q.-S.2003.Cellular automata models for synchronized traffic flow.J.Phys.A:Math.Gen.36:381-390.
[Jiang et al.2002]JIANG,R.,Wu,Q.-S.,AND ZHU,Z.-J.2002.A new continuum model for traffic flow and numerical tests.Transpn.Res.B 36:405-419.
[Jost and Nagel2002]JOST,D.AND NAGEL,K.2002.Probabilistic Traffic Flow Breakdown In Stochastic Car Following Models.http://citeseer.ist.psu.edu/jost02probabilistic.html.
[Kerner2004]KERNER,B.2004.Three-phase traffic theory and highway capacity.Physica A 333:379-440.
[Kerner and Klenov2002]KERNER,B.AND KLENOV,S.2002.A microscopic model for phase transitions in traffic flow.J.Phys.A:Math.Gen 35:L31-L43.
[Kerner and Klenov2003]KERNER,B.AND KLENOV,S.2003.Microscopic theory of spatial-temporal congested traffic patterns at highway bottlenecks.Phys.Rev.E 68:036130(20).
[Kerner et al.2002]KERNER,B.,KLENOV,S.,AND WOLF,D.2002.Cellular automata approach to three-phase traffic theory.J.Phys.A:Math.Gen.35:9971-10013.
[Kerner and Konhauser1993]KERNER,B.AND KONHAUSER,P.1993.Cluster effect in initially homogeneous traffic flow.Phys.Rev.E 48:2335(4).
[Kerner and Rehborn1996]KERNER,B.AND REHBORN,H.1996.Experimental properties of complexity in traffic flow.Phys.Rev.E 53:4275(4).
[Kerner and Rehborn1997]KERNER,B.AND REHBORN,H.1997.Experimental properties of phase transitions in traffic flow.Phys.Rev.Lett 79:4030-4033.
[Kerner2002a]KERNER,B.S.2002a.Empirical macroscopic features of spatialtemporal traffic patterns at highway bottlenecks.Phys.Rev.E 65:046138(30).
[Kerner2002b]KERNER,B.S.2002b.Synchronized flow as a new traffic phase and related problems for traffic flow modelling.Math.Comput.Modell.35:481-508.
[Kerner et al.2006]KERNER,B.S.,KLENOV,S.L.,AND HILLER,A.2006.Criterion for traffic phases in single vehicle data and empirical test of a microscopic three-phase traffic theory.J.Phys.A:Math.Gen.39:2001-2020.
[Klar and Wegener1997]KLAR,A.AND WEGENER,R.1997.Enskog-like kinetic models for vehicular traffic.J.Stat.Phys.87:91-114.
[Knospe et al.2000]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2000.Towards a realistic microscopic description of highway traffic.J.Phys.A:Math.Gen.33:L477-L485.
[Knospe et al.2002a]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2002a.A realistic two-lane traffic model for highway traffic.J.Phys.A:Math.Gen.35:3369-3388.
[Knospe et al.2002b]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2002b.Single-vehicle data of highway traffic:Microscopic description of traffic phases.Phys.Rev.E 65:056133-16.
[Knospe et al.2004]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2004.Empirical test for cellular automaton models of traffic flow.Phys.Rev.E 70:016115(25).
[Kuhne1984]KUHNE,R.1984.Macroscopic freeway model for dense traffic-stopstart waves and incident detection.In J.Volmuller and R.Hamerslag(eds.),Proc.9th Int.Symp.on Transp.and Traffic Theory.VNU Science Press.
[Lallemand and Luo2003]LALLEMAND,P.AND LUO,L.-S.2003.Lattice Boltzmann method for moving boundaries.J.Comput.Phys.1:406-421.
[Lee et al.2000]LEE,H.-Y.,LEE,H.-W.,AND KIM,D.2000.Phase diagram of congested traffic flow:An empirical study.Phys.Rev.E 62:4737-4741.
[Li et al.2003]LI,L.,SATO,Y.,AND ZHU,H.2003.Simulating spatial urban expansion based on a physical process.Landscape Urban Plann 64:67-76.
[Lighthill and Whitham1955]LIGHTHILL,M.J.AND WHITHAM,G.B.1955.On kinematic waves.Ⅱ.A theory of traffic flow on long crowded roads.Proc.R.Soc.London,Ser.A 229:317-345.
[Ling et al.2004]LING,L.,LI,W.-Q.,WANG,W.,AND XU,L.-J.2004.Analysis of traffic characteristics of mixed traffic flow of automobile and motorocycle in signalized intersection.Journal of Highway and Transportation Research and Development 21:0113-0116(in chinese).
[Lo Schiavo2002]Lo SCHIAVO,M.2002.A personalized kinetic model of traffic flow.Math.Comput.Modell.35:607-622.
[Lubashevsky et al.2002]LUBASHEVSKY,I.,MAHNKE,R.,WAGNER,P.,AND KALENKOV,S.2002.Long-lived states in synchronized traffic flow:Empirical prompt and dynamical trap model.Phys.Rev.E 66:016117(13).
[Maerivoe and Moor2005]MAERIVOE,S.AND MOOR,S.D.2005.Cellular automata models of road traffic.Phys.Rep 419:1-64.
[Matsuhashi et al.2005]MATSUHASHI,N.,HYODO,T.,AND TAKAHASHI,Y.2005.Image processing analysis of motorcycle oriented mixed traffic flow in vietnam.In Proceedings of the Eastern Asia Society for Transportation Studies,volume 5,pp.929-944.
[McNamara and Zanetti1988]MCNAMARA,G.R.AND ZANETTI,G.1988.Use of the Boltzmann equation to simulate lattice-gas automata.Phys.Rev.Lett.61:2332-2335.
[Meng et al.2008]MENG,J.-P.,QIAN,Y.-H.,LI,X.-L.,AND DAI,S.-Q.2008.Lattice Boltzmann model for traffic flow.Phys.Rev.E 77:036108(9).
[Nagatani1993]NAGATANI,T.1993.Effect of traffic accident on jamming transition in traffic-flow model.J.Phys.A:Math.Gen.26:L1015-L1020.
[Nagatani1999a]NAGATANI,T.1999a.Jamming transition in a two-dimensional traffic flow model.Phys.Rev.E 59:4857-4864.
[Nagatani1999b]NAGATANI,T.1999b.TDGL and MKDV equations for jamming transition in the lattice models of traffic.Physica A 264:581-592.
[Nagatani2000]NAGATANI,T.2000.Density waves in traffic flow.Phys.Rev.E 61:3564-3570.
[Nagatani2002]NAGATANI,T.2002.The physics of traffic jams.Rep.Prog.Phys.65:1331-1386.
[Nagel and Schreckenberg1992]NAGEL,K.AND SCHRECKENBERG,M.1992.A cellular automaton model for freeway traffic.J.Phys.I France 2:2221-2229.
[Nagel et al.1997]NAGEL,K.,STRETZ,P.,PIECK,M.,LECKEY,S.,DONNELLY,R.,AND BARRETT,C.L.1997.Transims traffic flow characteristics.Los Alamos Unclassified Report No.(LA-UR) 97-3530.
[Nagel et al.1998]NAGEL,K.,WOLF,D.,WAGNER,P.,AND SIMON,P.1998.Two-lane traffic rules for cellular automata:A systematic approach.Phys.Rev.E 58:1425-1437.
[Nelson2000]NELSON,P.2000.Synchronized traffic flow from a modified Lighthill-Whitman model.Phys.Rev.E 61:6052(4).
[Nelson and Sopasakis1998]NELSON,P.AND SOPASAKIS,A.1998.The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations.Transpn.Res.B 32:589-604.
[Neubertetal.1999]NEUBERT,L.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.1999.Single-vehicle data of highway traffic:A statistical analysis.Phys.Rev.E 60:6480-6490.
[Nishinari2001]NISHINARI,K.2001.A Lag-range representation of cellular automaton traffic-flow models.J.Phys.A:Math.Gen.34:10727-10736.
[Pipes1953]PIPES,L.1953.An operational analysis of traffic dynamics.J.Appl.Phys.24:274-281.
[Prigogine and Herman1971]PRIGOGINE,I.AND HERMAN,R.C.1971.Kinetic theory of vehiular traffic.American Elsevier Publishing.Co.
[Qian et al.1992]QIAN,Y.-H.,D'HUMIeRES,D.,AND LALLEMAND,P.1992.Lattice BGK model for Navier-Stokes equation.Europhys.Lett.17:479-484.
[Rascle2002]RASCLE,M.2002.An improved macroscopic model of traffic flow:Derivation and links with the Lighthill-Whitham model.Math.Comput.Modell.35:581-590.
[Reuschel1950]REUSCHEL,A.1950.Fahrzeugbewegungen in der kolonne.Oesterr.Ingr.Arch pp.193-215.
[Richards1956]RICHARDS,P.1956.Shock waves on the highway.Oper.Res.4:42-51.
[Rickert et al.1996]RICKERT,M.,NAGEL,K.,SCHRECKENBERG,M.,AND LATOUR,A.1996.Two lane traffic simulations using cellular automata.Physica A 231:534-550.
[Siebel and Mauser2006]SIEBEL,F.AND MAUSER,W.2006.Synchronized flow and wide moving jams from balanced vehicular traffic.Phys.Rev.E 73:066108(10).
[Sirakoulis et al.2003]SIRAKOULIS,G.,KARAFYLLIDIS,I.,MIZAS,C.,MARDIRIS,V.,THANAILAKIS,A.,AND TSALIDES,P.2003.A cellular automaton model for the study of DNA sequence evolution.Comput.Biol.Med.33:439-453.
[Sopasakis2002]SOPASAKIS,A.2002.Unstable flow theory and modeling.Math.Comput.Modell.35:623-641.
[Tadaki et al.2006]TADAKI,S.,KIKUCHI,M.,NAKAYAMA,A.,NISHINARI,K.,SHIBATA,A.,SUGIYAMA,Y.,AND YUKAWA,S.2006.Power-law fluctuation in expressway traffic flow:Detrended fluctuation analysis.J.Phys.Soc.Jpn 75:034002.
[Takayasu and Takayasu1993]TAKAYASU,M.AND TAKAYASU,H.1993.1/f noise in a traffic model.Fractals 1:860-866.
[Tanaka-Yamawaki etal.1996]TANAKA-YAMAWAKI,M.,KITAMIKADO,S.,AND FUKUDA,T.1996.Consensus formation and the cellular automata.Robotics and Autonomous Systems 19:15-22.
[Treiber et al.1999]TREIBER,M.,HENNECKE,A.,AND HELBING,D.1999.Derivation,properties,and simulation of a gas-kinetic-based,nonlocal traffic model.Phys.Rev.E 59:239-253.
[Treiber et al.2000]TREIBER,M.,HENNECKE,A.,AND HELBING,D.2000.Congested traffic states in empirical observations and microscopic simulations.Phys.Rev.E 62:1805-1824.
[Velasco and W.Marques2005]VELASCO,R.M.AND W.MARQUES,J.2005.Navier-Stokes-like equations for traffic flow.Phys.Rev.E 72:046102(9).
[Wagner et al.1997]WAGNER,P.,NAGEL,K.,AND WOLF,D.1997.Realistic multi-lane traffic rules for cellular automata.Physica A 234:687-698.
[Wahle et al.2000]WAHLE,J.,BAZZAN,A.L.C.,KLUGL,F.,AND SCHRECKENBERG,M.2000.Decision dynamics in a traffic scenario.Physica A 287:669-681.
[Wahle et al.2002]WAHLE,J.,CHROBOK,R.,POTTMEIER,A.,AND SCHRECKENBERG,M.2002.A microscopic simulator for freeway traffic.Networks and Spatial Economics 2:371-386.
[Waldeer2003]WALDEER,K.2003.The direct simulation Monte Carlo method applied to a Boltzmann-like vehicular traffic flow model.Comput.Phys.Commun.156:1-12.
[Wang et al.1998b]WANG,B.-H.,KWONG,Y.-R.,AND HUI,P.-M.1998b.Statistical mechanical approach to Fukui-Ishibashi traffic flow models.Phys.Rev.E 57:2568-2573.
[Wang et al.2005]WANG,W.-X.,WANG,B.-H.,ZHENG,W.-C.,YIN,C.-Y.,AND ZHOU,T.2005.Advanced information feedback in intelligent traffic systems.Phys.Rev.E 72:066702(6).
[Whitham1974]WHITHAM,G.1974.Linear and Nonlinear Waves.John Wiley &Sons.
[Wolf-Gladrow2000]WOLF-GLADROW,D.2000.Lattice-gas cellular automata and lattice Boltzmann models:An introduction.Number 1725 in Lecture Notes in Mathematics.Springer.
[Wolfram1986]WOLFRAM,S.1986.Cellular automaton fluids 1:Basic theory.J.Star.Phys.45:471-526.
[Xue et al.2002]XUE,Y.,DAI,S.-Q.,AND GU,G.-Q.2002.Analysis of phase transition of traffic flow in a two-layer network.In W.-Z.Chien(ed.),ICNM-IV,pp.913-918.Shanghai University Press.
[Xue et al.2001]XUE,Y.,DONG,L.-Y.,AND DAI,S.-Q.2001.Study on onedimensional model of traffic flow with stochastic deceleration via lattice Boltzmann method.Journal of Shanghai University(English Edition) 5:104-106.
[Yan and Hu2000]YAN,G.-W.AND HU,S.-X.2000.Simple lattice Boltzmann model for traffic flows.Acta Mechanica Sinica 16:70-74.
[Yokoya2004]YOKOYA,Y.2004.Dynamics of traffic flow with real-time traffic information.Phys.Rev.E69:016121(11).
[Zhang et al.2005]ZHANG,P.,LIU,R.-X.,AND WONG,S.-C.2005.Highresolution numerical approximation of traffic flow problems with variable lanes and free-flow velocities.Phys.Rev.E 71:056704(13).
[Zhang et al.2006]ZHANG,P.,WONG,S.-C.,AND SHU,C.-W.2006.A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway.J.Comput.Phys.212:739-756.
[戴世强＆薛郁2004]戴世强,薛郁.2004.交通流的建模与仿真,交通流与颗粒流(唐孝威等主编).浙江大学出版社.
[冯蔚东等2003]冯蔚东,陈剑,贺国光,刘豹.2003.交通流中的分形研究.高技术通讯pp.59-65.
[付传技等2006]付传技,汪秉宏,殷传洋,高坤.2006.利用智能决策的双通道交通流.物理学报.55:4032-4038.
[甘明星等2004]甘明星,张小军,刘祖源.2004.交通流的格子Boltzmann模型.武汉理工大学学报(交通科学与工程版)28:453-455.
[甘明星2004]甘明星.2004.基于Lattice Boltzmann方法的交通流模型研究.武汉理工大学硕士学位论文.
[黄海军2004]黄海军.2004.交通流理论的创立.流体中文网论坛-交通流论坛
[黄乒花等2002]黄乒花,孔令江,刘慕仁.2002.用格子Boltzmann方法研究交通流BML模型.广西师范大学学报(自然科学版)20:6-10.
[黄乒花2001]黄乒花.2001.用元胞自动机和格子玻尔兹曼方法研究交通流的有关问题.广西师范大学硕士学位论文.
[任福田等2003]任福田,刘小明,荣建.2003.交通工程学.人民交通出版社.
[宋涛2006]宋涛.2006.城市地面-高架路交通复杂网络的超越图模型及动力学模拟.上海大学硕士学位论文.
[SCCTPI2005]2005.上海城市综合交通发展报告.技术报告,上海市城市综合交通规划研究所.
[汪秉宏等1998a]汪秉宏,邝乐琪,许伯铭.1998a.高速公路交通流元胞自动机模型的一种统计平均解耦处理.物理学报47:906-915.
[张鹏等2005]张鹏,戴世强,刘儒勋.2005.多等级交通流LWR模型中的非线性波描述与WENO数值逼近.应用数学和力学26:637-644.
[Aw and Rascle2000]Aw,A.AND RASCLE,M.2000.Resurrection of "second order" models of traffic flow.SIAM J.Appl.Math.60:916-938.
[Bando et al.1995a]BANDO,M.,HASEBE,K.,NAKANISHI,K.,NAKAYAMA,A.,SHIBATA,A.,AND SUGIYAMA,Y.1995a.Phenomenological study of dynamical model of traffic flow.J.Phys.I France 5:1389-1399.
[Bando et al.1995b]BANDO,M.,HASEBE,K.,NAKAYAMA,A.,SHIBATA,A.,AND SUGIYAMANO,V.1995b.Dynamical model of traffic congestion and numerical simulation.Phys.Rev.E 51:1035-1042.
[Barlovic et al.1998]BARLOVIC,R.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.1998.Metastable states in cellular automata for traffic flow.Eur.Phy.J.B 5:793-800.
[Bellomo et al.2002]BELLOMO,N.,DELITALA,M.,AND COSCIA,V.2002.On the mathematical theory of vehicular traffic flow I:Fluid dynamic and kinetic modelling.Math.Models Meth.Appl.Sci.12:1801-1843.
[Benjamin et al.1996]BENJAMIN,S.C.,JOHNSON,N.F.,AND HUI,P.M.1996.Cellular automata models of traffic flow along a highway containing a junction.J.Phys.A:Math.Gen.29:3119-3127.
[Benzi et al.1992]BENZI,R.,SUCCI,S.,AND VERGASSOLA,M.1992.The lattice Boltzmann equation:Theory and applications.Rhys.Rep 222:145-197.
[Bhatnagar et al.1954]BHATNAGAR,P.,GROSS,E.,AND KROOK,M.1954.A model for collision processs in gases.I.Small amplitude processess in charged and neutral one-component system.Phys.Rev 94:511-525.
[Biham et al.1992]BIHAM,O.,MIDDLETON,A.,AND LEVINE,D.1992.Selforgranization and a dynamical transition in traffic-flow models.Phys.Rev.A 46:R6124-R6127.
[Buchanan2008]BUCHANAN,M.2008.Physics is dead,long live physics! Nature physics 4:159.
[Chandler et al.1958]CHANDLER,R.,HERMAN,R.,AND MONTROLL,E.1958.Traffic dynamics:Studies in car following.Oper.Res.6:165-184.
[Chen et al.1992]CHEN,H.-D.,CHEN,S.-V.,AND MATTHAEUS,W.H.1992.Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method.Phys.Rev.A 45:R5339-R5342.
[Chen et al.2004]CHEN,W.,HUANG,D.,HUANG,W.,AND HWANG,W.2004.Traffic flow on a 3-lane highway.Int.J.Mod.Phys.B 18:4161-4171.
[Cheybani et al.2000a]CHEYBANI,S.,KERTESZ,J.,AND SCHRECKENBERG,M.2000a.Nondeterministic Nagel-Schreckenberg traffic model with open boundary conditions.Phys.Rev.E 63:016108(9).
[Cheybani et al.2000b]CHEYBANI,S.,KERTESZ,J.,AND SCHRECKENBERG,M.2000b.Stochastic boundary conditions in the deterministic Nagel-Schreckenberg traffic model.Phys.Rev.E 63:016107.
[Cho and Lo2002]CHO,H.-J.AND LO,S.-C.2002.Modeling self-consistent multi-class dynamic traffic flow.Physica A 312:342-362.
[Chopard et al.1996]CHOPARD,B.,LUTHI,P.,AND QUELOZ,P.1996.Cellular automata model of car traffic in a two-dimensional street network.J.Phys.A:Math.and Gen.29:2325-2336.
[Chowdhury et al.2000]CHOWDHURY,D.,SANTEN,L.,AND SCHADSCHNEIDER,A.2000.Statistical physics of vehicular traffic and some realted systems.Phys.Rep 329:199-329.
[Chowdhury and Schadschneider1999]CHOWDHURY,D.AND SCHADSCHNEIDER,A.1999.Self-organization of traffic jams in cities:Effects of stochastic dynamics and signal periods.Phys.Rev.E 59:R1311-R1314.
[Chowdhury et al.1997]CHOWDHURY,D.,WOLF,D.,AND SCHRECKENBERG,M.1997.Particle hopping models for two-lane traffic with two kinds of vehicles:Effects of lane-changing rules.Physica A 235:417-439.
[Chung et al.1995]CHUNG,K.-H.,HUI,P.-M.,AND Gu,G.-Q.1995.Twodimensional traffic flow problems with faulty traffic lights.Phys.Rev.E 51:772-774.
[Coclite et al.2005]COCLITE,G.,GARAVELLO,M.,AND PICCOLI,B.2005.Traffic flow on a road network.SIAM J.Math.Anal.36:1862-1886.
[Coscia et al.2007]COSCIA,V.,DELITALA,M.,AND FRASCA,P.2007.On the mathematical theory of vehicular traffic flow Ⅱ:Discrete velocity kinetic models.Int.J.Non-Linear Mech.42:411-421.
[Daganzo1995]DAGANZO,C.1995.Requiem for second-order fluid approximations of traffic flow.Transpn.Res.B 29B:277-286.
[Daganzo1994]DAGANZO,C.F.1994.The cell transmission model:A dynamic representation of highway traffic consistent with the hydrodynamic theory.Transpn.Res.B 28:269-287.
[Delitala2003]DELITALA,M.2003.Nonlinear models of vehicular traffic flownew frameworks of the mathematical kinetic theory.Comptes Rendus Mecanique 331:817-822.
[Ebersbach and Schneider2004]EBERSBACH,A.AND SCHNEIDER,J.2004.Twolane traffic with places of obstruction to traffic.Int.J.Mod.Phys.C 15:535-544.
[Ebersbach et al.2000]EBERSBACH,A.,SCHNEIDER,J.,MORGENSTERN,I.,AND HAMMWOHNER,R.2000.The influence of trucks on traffic flow-an investigation on the Nagel-Schreckenberg-model.Int.J.Mod.Phys.C 11:837-842.
[Ez-Zahraouya et al.2004]EZ-ZAHRAOUYA,H.,JETTO,K.,AND BENYOUSSEF,A.2004.The effect of mixture lengths of vehicles on the traffic flow behaviour in one-dimensional cellular automaton.Eur.Phys.J.B 40:111-117.
[Feng et al.1997]FENG,S.-W.,Gu,G.-Q.,AND DAI,S.-Q.1997.Effects of traffic lights on CA traffic model.Communications in Nonlinear Science and Numerical Simulation 2:70-74.
[Frisch et al.1986]FRISCH,U.,HASSLACHER,B.,AND POMEAU,Y.1986.Lattice-gas automata for the Navier-Stokes equation.Phys.Rev.Lett.56:1505-1508.
[Fukui and Ishibashi1996]FUKUI,M.AND ISHIBASHI,Y.1996.Traffic flow in 1D cellular automaton model including cars moving with high speed.J.Phys.Soc.Jpn 65:1868-1870.
[Garavello and Piccoli2006]GARAVELLO,M.AND PICCOLO,B.2006.Traffic Flow on Networks,volume 1 of AIMS Series on Applied Math.American Institute of Mathematical Sciences.
[Gartner et al.1999]N.Gartner,C.Messer,and A.Rathi(eds.) 1999.Revised monograph on traffic flow theory.The federal highway administration(FHWA).
[Gazis2002]GAZIS,D.2002.The origins of traffic theory.Oper.Res.50:69-77.
[Ge et al.2004]GE,H.-X.,DAI,S.-Q.,DONG,L.-Y.,AND XUE,Y.2004.Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application.Phys.Rev.E 70:066134.
[Gu et al.1995]Gu,G.-Q.,CHUNG,K.-H.,AND HUI,P.-M.1995.Twodimensional traffic flow problems in inhomogeneous lattices.Physica A 217:339-347.
[He and Luo1997a]HE,X.-Y.AND LUO,L.-S.1997a.A priori derivation of the lattice Boltzmann equation.Phys.Rev.E 55:6333(4).
[He and Luo1997b]HE,X.-Y.AND LUO,L.-S.1997b.Theory of the lattice Boltzmann method:from the Boltzmann equation to the lattice Boltzmann equation.Phys.Rev.E 56:6811-6817.
[Helbing1995]HELBING,D.1995.Improved fluid-dynamic model for vehicular traffic.Phys.Rev.E 51:3164-3169.
[Helbing1996]HELBING,D.1996.Gas-kinetic derivation of Navier-Stokes-like traffic equations.Phys.Rev.E 53:2366-2381.
[Helbing1997a]HELBING,D.1997a.Empirical traffic data and their implications for traffic modeling.Phys.Rev.E 55:25(4).
[Helbing1997b]HELBING,D.1997b.Modeling multi-lane traffic flow with queuing effects.Physica A 242:175-194.
[Helbing1998]HELBING,D.1998.From microscopic to macroscopic traffic models.arXiv:cond-mat/9806171v1.
[Helbing2001]HELBING,D.2001.Traffic and related self-driven many-particles systems.Rev.Mod.Phys 73:1067-1141.
[Helbing et al.2001]HELBING,D.,HENNECKE,A.,SHVETSOV,V.,AND TREIBER,M.2001.MASTER:macroscopic traffic simulation based on a gaskinetic,non-local traffic model.Transpn.Res.B 35:183-211.
[Helbing and Tilch1998]HELBING,D.AND TILCH,B.1998.Generalized force model of traffic dynamics.Phys.Rev.E 58:133-138.
[Holden and Risebro1995]HOLDEN,H.AND RISEBRO,N.H.1995.A mathematical model of traffic flow on a network of unidirectional roads.SIAM J.Math.Anal.26:999-1017.
[Hoogendoorn and Bovy2000]HOOGENDOORN,S.AND BOVY,P.2000.Continuum modeling of multiclass traffic flow.Transpn.Res.B 34:123-146.
[Hoogendoorn and Bovy2001]HOOGENDOORN,S.AND BOVY,P.2001.Generic gas-kinetic traffic systems modeling with applications to vehicular traffic flow.Transpn.Res.B 35:317-336.
[Hu et al.2003]Hu,T.-P.,AHMAD,F.,AND NGUYEN,X.-D.2003.A comparison study on motorcycle traffic development in some Asian countries-case of Taiwan,Malaysia and Vietnam.Technical report,Institute of Civil Engineering National Taiwan University.
[Huang and Huang2006]HUANG,D.-W.AND HUANG,W.-N.2006.Biham-Middleton-Levine model with four-directional traffic.Physica A 370:747-755.
[Jia et al.2004]JIA,B.,JIANG,R.,AND WU,Q.-S.2004.A realistic two-lane cellular automaton model for traffic flow.Int.J.Mod.Phys.C 15:381-392.
[Jiang and Wu2003]JIANG,R.AND WU,Q.-S.2003.Cellular automata models for synchronized traffic flow.J.Phys.A:Math.Gen.36:381-390.
[Jiang et al.2002]JIANG,R.,Wu,Q.-S.,AND ZHU,Z.-J.2002.A new continuum model for traffic flow and numerical tests.Transpn.Res.B 36:405-419.
[Jost and Nagel2002]JOST,D.AND NAGEL,K.2002.Probabilistic Traffic Flow Breakdown In Stochastic Car Following Models.http://citeseer.ist.psu.edu/jost02probabilistic.html.
[Kerner2004]KERNER,B.2004.Three-phase traffic theory and highway capacity.Physica A 333:379-440.
[Kerner and Klenov2002]KERNER,B.AND KLENOV,S.2002.A microscopic model for phase transitions in traffic flow.J.Phys.A:Math.Gen 35:L31-L43.
[Kerner and Klenov2003]KERNER,B.AND KLENOV,S.2003.Microscopic theory of spatial-temporal congested traffic patterns at highway bottlenecks.Phys.Rev.E 68:036130(20).
[Kerner et al.2002]KERNER,B.,KLENOV,S.,AND WOLF,D.2002.Cellular automata approach to three-phase traffic theory.J.Phys.A:Math.Gen.35:9971-10013.
[Kerner and Konhauser1993]KERNER,B.AND KONHAUSER,P.1993.Cluster effect in initially homogeneous traffic flow.Phys.Rev.E 48:2335(4).
[Kerner and Rehborn1996]KERNER,B.AND REHBORN,H.1996.Experimental properties of complexity in traffic flow.Phys.Rev.E 53:4275(4).
[Kerner and Rehborn1997]KERNER,B.AND REHBORN,H.1997.Experimental properties of phase transitions in traffic flow.Phys.Rev.Lett 79:4030-4033.
[Kerner2002a]KERNER,B.S.2002a.Empirical macroscopic features of spatialtemporal traffic patterns at highway bottlenecks.Phys.Rev.E 65:046138(30).
[Kerner2002b]KERNER,B.S.2002b.Synchronized flow as a new traffic phase and related problems for traffic flow modelling.Math.Comput.Modell.35:481-508.
[Kerner et al.2006]KERNER,B.S.,KLENOV,S.L.,AND HILLER,A.2006.Criterion for traffic phases in single vehicle data and empirical test of a microscopic three-phase traffic theory.J.Phys.A:Math.Gen.39:2001-2020.
[Klar and Wegener1997]KLAR,A.AND WEGENER,R.1997.Enskog-like kinetic models for vehicular traffic.J.Stat.Phys.87:91-114.
[Knospe et al.2000]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2000.Towards a realistic microscopic description of highway traffic.J.Phys.A:Math.Gen.33:L477-L485.
[Knospe et al.2002a]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2002a.A realistic two-lane traffic model for highway traffic.J.Phys.A:Math.Gen.35:3369-3388.
[Knospe et al.2002b]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2002b.Single-vehicle data of highway traffic:Microscopic description of traffic phases.Phys.Rev.E 65:056133-16.
[Knospe et al.2004]KNOSPE,W.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.2004.Empirical test for cellular automaton models of traffic flow.Phys.Rev.E 70:016115(25).
[Kuhne1984]KUHNE,R.1984.Macroscopic freeway model for dense traffic-stopstart waves and incident detection.In J.Volmuller and R.Hamerslag(eds.),Proc.9th Int.Symp.on Transp.and Traffic Theory.VNU Science Press.
[Lallemand and Luo2003]LALLEMAND,P.AND LUO,L.-S.2003.Lattice Boltzmann method for moving boundaries.J.Comput.Phys.1:406-421.
[Lee et al.2000]LEE,H.-Y.,LEE,H.-W.,AND KIM,D.2000.Phase diagram of congested traffic flow:An empirical study.Phys.Rev.E 62:4737-4741.
[Li et al.2003]LI,L.,SATO,Y.,AND ZHU,H.2003.Simulating spatial urban expansion based on a physical process.Landscape Urban Plann 64:67-76.
[Lighthill and Whitham1955]LIGHTHILL,M.J.AND WHITHAM,G.B.1955.On kinematic waves.Ⅱ.A theory of traffic flow on long crowded roads.Proc.R.Soc.London,Ser.A 229:317-345.
[Ling et al.2004]LING,L.,LI,W.-Q.,WANG,W.,AND XU,L.-J.2004.Analysis of traffic characteristics of mixed traffic flow of automobile and motorocycle in signalized intersection.Journal of Highway and Transportation Research and Development 21:0113-0116(in chinese).
[Lo Schiavo2002]Lo SCHIAVO,M.2002.A personalized kinetic model of traffic flow.Math.Comput.Modell.35:607-622.
[Lubashevsky et al.2002]LUBASHEVSKY,I.,MAHNKE,R.,WAGNER,P.,AND KALENKOV,S.2002.Long-lived states in synchronized traffic flow:Empirical prompt and dynamical trap model.Phys.Rev.E 66:016117(13).
[Maerivoe and Moor2005]MAERIVOE,S.AND MOOR,S.D.2005.Cellular automata models of road traffic.Phys.Rep 419:1-64.
[Matsuhashi et al.2005]MATSUHASHI,N.,HYODO,T.,AND TAKAHASHI,Y.2005.Image processing analysis of motorcycle oriented mixed traffic flow in vietnam.In Proceedings of the Eastern Asia Society for Transportation Studies,volume 5,pp.929-944.
[McNamara and Zanetti1988]MCNAMARA,G.R.AND ZANETTI,G.1988.Use of the Boltzmann equation to simulate lattice-gas automata.Phys.Rev.Lett.61:2332-2335.
[Meng et al.2008]MENG,J.-P.,QIAN,Y.-H.,LI,X.-L.,AND DAI,S.-Q.2008.Lattice Boltzmann model for traffic flow.Phys.Rev.E 77:036108(9).
[Nagatani1993]NAGATANI,T.1993.Effect of traffic accident on jamming transition in traffic-flow model.J.Phys.A:Math.Gen.26:L1015-L1020.
[Nagatani1999a]NAGATANI,T.1999a.Jamming transition in a two-dimensional traffic flow model.Phys.Rev.E 59:4857-4864.
[Nagatani1999b]NAGATANI,T.1999b.TDGL and MKDV equations for jamming transition in the lattice models of traffic.Physica A 264:581-592.
[Nagatani2000]NAGATANI,T.2000.Density waves in traffic flow.Phys.Rev.E 61:3564-3570.
[Nagatani2002]NAGATANI,T.2002.The physics of traffic jams.Rep.Prog.Phys.65:1331-1386.
[Nagel and Schreckenberg1992]NAGEL,K.AND SCHRECKENBERG,M.1992.A cellular automaton model for freeway traffic.J.Phys.I France 2:2221-2229.
[Nagel et al.1997]NAGEL,K.,STRETZ,P.,PIECK,M.,LECKEY,S.,DONNELLY,R.,AND BARRETT,C.L.1997.Transims traffic flow characteristics.Los Alamos Unclassified Report No.(LA-UR) 97-3530.
[Nagel et al.1998]NAGEL,K.,WOLF,D.,WAGNER,P.,AND SIMON,P.1998.Two-lane traffic rules for cellular automata:A systematic approach.Phys.Rev.E 58:1425-1437.
[Nelson2000]NELSON,P.2000.Synchronized traffic flow from a modified Lighthill-Whitman model.Phys.Rev.E 61:6052(4).
[Nelson and Sopasakis1998]NELSON,P.AND SOPASAKIS,A.1998.The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations.Transpn.Res.B 32:589-604.
[Neubertetal.1999]NEUBERT,L.,SANTEN,L.,SCHADSCHNEIDER,A.,AND SCHRECKENBERG,M.1999.Single-vehicle data of highway traffic:A statistical analysis.Phys.Rev.E 60:6480-6490.
[Nishinari2001]NISHINARI,K.2001.A Lag-range representation of cellular automaton traffic-flow models.J.Phys.A:Math.Gen.34:10727-10736.
[Pipes1953]PIPES,L.1953.An operational analysis of traffic dynamics.J.Appl.Phys.24:274-281.
[Prigogine and Herman1971]PRIGOGINE,I.AND HERMAN,R.C.1971.Kinetic theory of vehiular traffic.American Elsevier Publishing.Co.
[Qian et al.1992]QIAN,Y.-H.,D'HUMIeRES,D.,AND LALLEMAND,P.1992.Lattice BGK model for Navier-Stokes equation.Europhys.Lett.17:479-484.
[Rascle2002]RASCLE,M.2002.An improved macroscopic model of traffic flow:Derivation and links with the Lighthill-Whitham model.Math.Comput.Modell.35:581-590.
[Reuschel1950]REUSCHEL,A.1950.Fahrzeugbewegungen in der kolonne.Oesterr.Ingr.Arch pp.193-215.
[Richards1956]RICHARDS,P.1956.Shock waves on the highway.Oper.Res.4:42-51.
[Rickert et al.1996]RICKERT,M.,NAGEL,K.,SCHRECKENBERG,M.,AND LATOUR,A.1996.Two lane traffic simulations using cellular automata.Physica A 231:534-550.
[Siebel and Mauser2006]SIEBEL,F.AND MAUSER,W.2006.Synchronized flow and wide moving jams from balanced vehicular traffic.Phys.Rev.E 73:066108(10).
[Sirakoulis et al.2003]SIRAKOULIS,G.,KARAFYLLIDIS,I.,MIZAS,C.,MARDIRIS,V.,THANAILAKIS,A.,AND TSALIDES,P.2003.A cellular automaton model for the study of DNA sequence evolution.Comput.Biol.Med.33:439-453.
[Sopasakis2002]SOPASAKIS,A.2002.Unstable flow theory and modeling.Math.Comput.Modell.35:623-641.
[Tadaki et al.2006]TADAKI,S.,KIKUCHI,M.,NAKAYAMA,A.,NISHINARI,K.,SHIBATA,A.,SUGIYAMA,Y.,AND YUKAWA,S.2006.Power-law fluctuation in expressway traffic flow:Detrended fluctuation analysis.J.Phys.Soc.Jpn 75:034002.
[Takayasu and Takayasu1993]TAKAYASU,M.AND TAKAYASU,H.1993.1/f noise in a traffic model.Fractals 1:860-866.
[Tanaka-Yamawaki etal.1996]TANAKA-YAMAWAKI,M.,KITAMIKADO,S.,AND FUKUDA,T.1996.Consensus formation and the cellular automata.Robotics and Autonomous Systems 19:15-22.
[Treiber et al.1999]TREIBER,M.,HENNECKE,A.,AND HELBING,D.1999.Derivation,properties,and simulation of a gas-kinetic-based,nonlocal traffic model.Phys.Rev.E 59:239-253.
[Treiber et al.2000]TREIBER,M.,HENNECKE,A.,AND HELBING,D.2000.Congested traffic states in empirical observations and microscopic simulations.Phys.Rev.E 62:1805-1824.
[Velasco and W.Marques2005]VELASCO,R.M.AND W.MARQUES,J.2005.Navier-Stokes-like equations for traffic flow.Phys.Rev.E 72:046102(9).
[Wagner et al.1997]WAGNER,P.,NAGEL,K.,AND WOLF,D.1997.Realistic multi-lane traffic rules for cellular automata.Physica A 234:687-698.
[Wahle et al.2000]WAHLE,J.,BAZZAN,A.L.C.,KLUGL,F.,AND SCHRECKENBERG,M.2000.Decision dynamics in a traffic scenario.Physica A 287:669-681.
[Wahle et al.2002]WAHLE,J.,CHROBOK,R.,POTTMEIER,A.,AND SCHRECKENBERG,M.2002.A microscopic simulator for freeway traffic.Networks and Spatial Economics 2:371-386.
[Waldeer2003]WALDEER,K.2003.The direct simulation Monte Carlo method applied to a Boltzmann-like vehicular traffic flow model.Comput.Phys.Commun.156:1-12.
[Wang et al.1998b]WANG,B.-H.,KWONG,Y.-R.,AND HUI,P.-M.1998b.Statistical mechanical approach to Fukui-Ishibashi traffic flow models.Phys.Rev.E 57:2568-2573.
[Wang et al.2005]WANG,W.-X.,WANG,B.-H.,ZHENG,W.-C.,YIN,C.-Y.,AND ZHOU,T.2005.Advanced information feedback in intelligent traffic systems.Phys.Rev.E 72:066702(6).
[Whitham1974]WHITHAM,G.1974.Linear and Nonlinear Waves.John Wiley &Sons.
[Wolf-Gladrow2000]WOLF-GLADROW,D.2000.Lattice-gas cellular automata and lattice Boltzmann models:An introduction.Number 1725 in Lecture Notes in Mathematics.Springer.
[Wolfram1986]WOLFRAM,S.1986.Cellular automaton fluids 1:Basic theory.J.Star.Phys.45:471-526.
[Xue et al.2002]XUE,Y.,DAI,S.-Q.,AND GU,G.-Q.2002.Analysis of phase transition of traffic flow in a two-layer network.In W.-Z.Chien(ed.),ICNM-IV,pp.913-918.Shanghai University Press.
[Xue et al.2001]XUE,Y.,DONG,L.-Y.,AND DAI,S.-Q.2001.Study on onedimensional model of traffic flow with stochastic deceleration via lattice Boltzmann method.Journal of Shanghai University(English Edition) 5:104-106.
[Yan and Hu2000]YAN,G.-W.AND HU,S.-X.2000.Simple lattice Boltzmann model for traffic flows.Acta Mechanica Sinica 16:70-74.
[Yokoya2004]YOKOYA,Y.2004.Dynamics of traffic flow with real-time traffic information.Phys.Rev.E69:016121(11).
[Zhang et al.2005]ZHANG,P.,LIU,R.-X.,AND WONG,S.-C.2005.Highresolution numerical approximation of traffic flow problems with variable lanes and free-flow velocities.Phys.Rev.E 71:056704(13).
[Zhang et al.2006]ZHANG,P.,WONG,S.-C.,AND SHU,C.-W.2006.A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway.J.Comput.Phys.212:739-756.