Kinetic Models on Networks with Applications Traffic Flow and Supply Chains

具有应用流量和供应链的网络动力学模型

• 批准号: 79828029
• 负责人: Professor Dr. Michael Herty
• 金额: --
• 依托单位: Lehr- und Forschungsgebiet Mathematik - Kontinuierliche Optimierung
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79828029?language=en
• 关键词: Kinetic; Models; Networks; Applications; Traffic

Kinetic equations can provide a fine-scale description of a variety of physical processes. We are interested in physical systems where an additional network structure is present. Such systems have been studied in the context of traffic and production networks on a coarse-scale only. Coarse scale or macroscopic models are based on partial differential equations (PDEs) for averaged quantities whereas fine scale models are based on PDEs for distribution functions. Considering a network topology many processes on different arcs have to be coupled by suitable coupling conditions. In the case of macroscopic models their detailed form is a point of ongoing discussion. In this proposal we start from a kinetic description of Boltzmann type with applications in traffic flow, supply chains as well as gas dynamics. The kinetic equations describe the dynamics on the arcs on the finest possible scale. Coupling conditions for the dynamics at vertices arise naturally due to the linear structure of these equations. Besides the analysis of these coupled systems we are interested in the coupling conditions obtained by introducing averaged quantities at the vertex using moment closure relations. This introduces possible nonlinear macroscopic coupling conditions. We compare these newly obtained conditions with already known coupling conditions for the macroscopic models. Additionally, we include discrete decisions in the coupling conditions. These decisions can model for example instantaneous switching between different modes of operation or planning and design decisions. For the simulation of networks governed by kinetic models, coupling conditions and possibly discrete decisionSj we introduce an appropriate formulation by discretization and reformulation as mixed-integer programming problems. We give numerical results for traffic flow and supply chain applications and compare these results with existing macroscopic approaches. In particular, we discuss a multi-scale approach combining parts of the production process with highly nonlinear behavior and classical discretization concepts with the mixed-integer formulation.

动力学方程可以提供各种物理过程的精细描述。我们感兴趣的物理系统中存在一个额外的网络结构。此类系统仅在粗略规模的交通和生产网络背景下进行了研究。粗尺度或宏观模型是基于偏微分方程（PDE）的平均量，而细尺度模型是基于偏微分方程的分布函数。考虑到网络的拓扑结构，不同弧上的许多过程必须通过适当的耦合条件来耦合。在宏观模型的情况下，它们的详细形式是一个正在讨论的问题。在这个建议中，我们从玻尔兹曼类型的动力学描述开始，在交通流，供应链以及气体动力学中的应用。动力学方程以尽可能精细的尺度描述了弧上的动力学。由于这些方程的线性结构，顶点处的动力学的耦合条件自然出现。除了这些耦合系统的分析，我们感兴趣的耦合条件，通过引入平均量在顶点使用矩封闭关系。这引入了可能的非线性宏观耦合条件。我们比较这些新获得的条件与已知的宏观模型的耦合条件。此外，我们还在耦合条件中加入了离散决策。这些决策可以对不同操作模式或规划和设计决策之间的瞬时切换进行建模。对于动力学模型，耦合条件和可能的离散decisionSj网络的模拟，我们引入了一个适当的配方离散化和重新制定的混合整数规划问题。我们给出了交通流和供应链应用的数值结果，并将这些结果与现有的宏观方法进行比较。特别是，我们讨论了一个多尺度的方法相结合的部分生产过程中的高度非线性行为和经典的离散化概念与混合整数制定。