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。我们对存在额外网络结构的物理系统感兴趣。此类系统仅在交通和生产网络的背景下进行了粗略的研究。粗尺度或宏观模型基于平均量的偏微分方程 (PDE)，而精细尺度模型基于分布函数的偏微分方程。考虑到网络拓扑，不同弧上的许多进程必须通过适当的耦合条件进行耦合。就宏观模型而言，其详细形式是持续讨论的焦点。在这个提案中，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得到应用。动力学方程以尽可能精细的尺度描述了电弧的动力学。由于这些方程的线性结构，顶点动力学的耦合条件自然出现。除了对这些耦合系统的分析之外，我们还对通过使用矩闭合关系在顶点引入平均量而获得的耦合条件感兴趣。这引入了可能的非线性宏观耦合条件。我们将这些新获得的条件与宏观模型已知的耦合条件进行比较。此外，我们在耦合条件中包含离散决策。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。为了模拟由动力学模型、耦合条件和可能的离散决策控制的网络，我们通过离散化和重新表述为混合整数规划问题引入了适当的表述。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了比较。特别是，我们讨论了一种多尺度方法，将部分生产过程与高度非线性行为和经典离散化概念与混合整数公式相结合。