Kinetic Models on Networks with Applications Traffic Flow and Supply Chains

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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.