城市快速路交通流模型的非平凡稳态解与拥堵交通

Frequent traffic jams make the situation of vehicular traffic severe in the urban of China. The steady-state solutions of the traffic flow model in urban freeway are investigated under the Lagrange coordinate in this project. The weak solution theory, the qualitative theory of differential equation and kinetic wave theory are applied to study the mechanism and characteristics of the congested traffic. We analyze the similarities and differences of the congested traffic patterns between the isotropic and anisotropic traffic flow models, and discuss the influence about the selected conservative solution variables. Through the numerical simulation, we derive several steady-state solutions which include the wide moving jam，narrow cluster, limit cycle, limit-spiral, saddle-limit cycle and saddle-spiral solutions. These solutions provide good explanations to the phenomena of the stop-and-go, oscillatory and homogeneous congestions in the real-world traffic. The bottleneck effects are investigated with the inhomogeneous road conditions. We also consider the instability caused by the higher-order effect in the model. The numerical results reproduce several traffic phenomena which include the stationary flow, oscillatory congested traffic and saturated flux platform. This project investigates the traffic flow model under the Lagrange coordinate, which makes the macro and micro models be compatible. Therefore, the theories of these two models could be developed in parallel. The studies about the mechanism of the congested traffic and the bottleneck effects are helpful to find the congestion control methods to relieve the congestion of the real-world traffic.

我国城市车辆交通现状形势严峻，交通拥堵频发。本项目基于Lagrange坐标，探讨城市快速路交通流模型的稳态解。运用微分方程弱解理论、定性理论和运动学波理论，研究拥挤交通的形成机理及其特性。讨论各向同性和各向异性交通流模型形成的多种拥挤交通模式的异同，分析模型守恒形式对其影响程度。借助数值模拟，得到模型的宽移动堵塞解、窄幅集簇解、极限环解、极限环—焦点解、鞍点—极限环解和鞍点—焦点解等稳态解，较好地解释现实交通中的停停走走、振荡拥挤和均匀拥挤等现象。讨论由非均匀道路引发的交通瓶颈效应，分析模型中高阶效应导致的不稳定，并通过数值模拟再现定常流、振荡拥挤交通以及饱和流量平台等现象。本项目基于Lagrange坐标研究交通流模型，实现宏观和微观模型的相容以及两类模型研究理论的联通；多种拥挤模式形成机理与瓶颈效应的分析，有助于拥堵控制和疏导方法的制定，从而缓解现实中的交通拥堵。

运用微分方程弱解理论、定性理论、渐近分析理论和运动学波理论，研究拥挤交通的形成机理及其特性。通过对模型进行非线性分析，着重讨论宏观高阶加粘模型的宽移动堵塞解和孤立波解，研究特征参数及守恒量对模型性态的影响。分析各向异性交通流模型在不同守恒形式下演化形成的多种拥挤交通模式的异同。借助数值模拟，得到模型的宽移动堵塞解、窄幅集簇解、极限环解、极限环—焦点解、鞍点—极限环解和鞍点—焦点解等稳态解，较好地解释现实交通中的停停走走、振荡拥挤和均匀拥挤等现象。讨论有坡道路引发的交通瓶颈效应，推导出瓶颈前排队长度和饱和流量平台临界密度的解析式，并通过数值模拟再现定常流、振荡拥挤交通以及饱和流量平台等现象。该研究工作能够实现宏观和微观模型的相容以及两类模型研究理论的联通，有助于拥堵控制和疏导方法的制定，从而缓解现实中的交通拥堵。