基于Riemann问题的交通网络流体力学建模与数值求解

Dynamic traffic assignment (DTA) problems serve as a theoretical fundamental for the establishment of the intelligent transportation system (ITS). For the passing decade, the cell-transmission (CT) model has gradually replaced the TRANSYT model for the simulation of traffic flow in DTA problems, because of its ability to describe the propagation of queue within a road section or across a junction. However, as a discrete version of the LWR model, the CT model is only able to describe the equilibrium traffic phase. That is, it simply assumes a fixed velocity-density curve in the phase plane, which does not well agree with the observation. Actually, there have been more advanced models proposed, such as the higher-order model and the multi-class model, which better depict traffic flow phenomena. Nevertheless, currently these models are only able to simulate traffic flow on a single road and therefore cannot be directly used to solve DTA problems which involve a traffic network, unless the boundary conditions at a node or junction for the homogeneous equations of these models are appropriately derived. Here, the boundary conditions refer to the numerical fluxes, which are essential to the design of a numercal scheme for soving the network. Solution for such boundary conditions gives rise to a Junction Riemann (JR) problem.. In this context, the proposed research project focuses on the JR problems by using a conserved high-order (CHO) model and a multi-class LWR (MCLWR) model, respectively. The problems are challenging because their solution requires a complex coupling at the junction between the "demands" in the incoming roads and the "supplies" in the outgoing roads, which is subject to the mathematical and physical properties of the models, and by which the outflows and inflows as the Riemann solvers would be assigned. For the CHO model that is composed of two equations, the exact Riemann solver， which is also known as the Godunov flux could hopefully be derived by adopting a Riemann invariant and taking the advantage of the model's consistency with the LWR model. For the MCLWR model that is composed of more equations, the eigen-structure cannot be obtained explicitly, hence it is only possible to derive the approximate Riemann solvers through wave propagation analysis.. The extended CHO model would be able to describe the instability of traffic flow and the resultant propagation of stop-and-go waves on the network. The extended MCLWR model reasonably divides vehicles into several types and thus could simulate overtake behavior on the network. Hence, the resultant DTA models would be more theoretically robust in solving a wide range of issues, which involve the signal control, reliability and sensitivity analysis, etc. Since the studied JR problems are the extension of the classic Riemann problem in hyperbolic conservation laws, the subject is also of much significance for the enrichment of knowledge in the relevant study fields.

通过研究守恒高阶(CHO)模型和多车种LWR模型的交叉口Riemann问题，得到在交叉口处CHO模型齐次方程的准确Riemann解或Godunov型数值流通量，以及多车种LWR模型的Riemann近似解或近似数值流通量，从而可以构造这两类模型求解网络交通流的一阶和高阶数值格式。将上述两类模型由目前只能求解路段推广到能求解网络，其主要意义是使它们能够应用于求解动态交通分配问题。由于CHO模型能够合理描述交通流不稳定现象，如时停时走波；多车种LWR模型可描述不同车型混流和超车，上述推广应用将改进现有动态交通分配模型的合理性和适用性，有力推动涵盖极广的相关问题，如信号灯控制、交通系统可靠性分析和敏感性分析等方面的研究进展，并为智能交通系统的建设和管理提供理论依据。由于交叉口Riemann问题是经典Riemann问题的推广，课题的开展还将丰富计算流体力学和守恒律方程理论的研究内容。

项目通过将适于路段的交通流宏观模型推广到交通网络，研究城市交通路网的交通流建模和演化问题。其关键是求解模型方程在交叉口处Riemann问题，得到各路段在交叉口处的流量分配，即关于微分方程的边界条件，从而可通过数值方法模拟交通流在网络上的演化。项目的研究内容还包括模型方程的解析性质研究，高阶格式构造，基于均衡或优化原理的二维城市交通演化，以及行人流建模和仿真等多个方面。. 基于CHO（Conserved Higher-Order）模型与经典LWR模型的相容性，我们利用Riemann不变量，巧妙求解了CHO模型的Riemann问题。同时推广了在路段上数值求解CHO模型的Godunov, Lax-Friedrichs, Enquist-Ohser和TF（Traffic Flow）数值流通量，并据此推广了求解模型方程的间断Galerkin格式。针对简单交叉口或流通量间断问题，构造了MCLWR（Multi-class LWR）模型的Delta映射算法与间断Galerkin数值格式。此外，还分别运用泰勒展开法和约化摄动法研究了一般高阶模型的近似孤立波解。. 项目将用户最优、预测用户最优和系统最优等均衡原理与二维的网络交通流模型结合，求解包括多个CBD的城市动态交通分配问题，所得结果可应用于城市的交通流日常需求分析与宏观规划及其控制。并基于用户最优原理提出了微观的多粒子模型和元胞自动机模型，用于描述行人流的瓶颈前成拱、对流成行和高密度挤压等现象。. 通过开发交通流仿真平台，初步实现了数值模拟的可视化实时仿真演化。尤其是，通过GPS数据提取北京市路网信息，应用推广的CHO路网模型模拟了首都机场高速和北京市主要路网的交通流演化。我们正在开展与有关部门合作，争取获得数据和项目支持，将仿真平台首先应用于北京市的城市交通流分析、预测和控制。