A mathematical model describing supply chains on a network is introduced. In particular, conditions on each vertex of the network are specified. Finally, this leads to a system of nonlinear conservation laws coupled to ordinary differential equations. To prove the existence of a solution we make use of the front tracking method. A comparison to another approach is given and numerical results are presented.

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We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide L ∞ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

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In this work, we extend the one-dimensional Keller–Segel model for chemotaxis to general network topologies. We define appropriate coupling conditions ensuring the conservation of mass and show the existence and uniqueness of the solution. For our computational studies, we use a positive preserving first-order scheme satisfying a network CFL condition. Finally, we numerically validate the Keller–Segel network model and present results regarding special network geometries.

Material flow simulation is in increasing need of multi-scale models. On the one hand, macroscopic flow models are used for large scale simulations with a large number of parts. On the other hand microscopic models are needed to describe the details of the production process. In this paper we present a hierarchy of models for material flow problems ranging from detailed microscopic, Discrete Element Method (DEM) type, models to macroscopic models using scalar conservation laws with nonlocal interaction terms. Numerical simulations are presented on all levels of the hierarchy and the results are compared to each other for several test cases.