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In this subproject we want to investigate a new approach for using transport metrics in networks, by constructing appropriate Wasserstein-type metrics. By defining suitable
dissipation functionals in the nodes and edges of the network a metric structure (ideally Riemannian) can be constructed in order to formulate different transport-diffusion equation with coupling conditions in the nodes as gradient flows. The instantaneous division of flows in the nodes as well as a (partial) storage will be investigated as different cases.

Based on gradient flow formulations we want to further advance the analysis of equations on networks as well as their discretization and numerical solution. In particular we expect important results on structure-preserving model reduction (via parameter asymptotics) and structure-preserving numerical solution of transport-diffusion equations on networks. In addition we want to investigate a scaling limit (quasi-stationary situations in edges) to algebraic or ordinary differential equations in the nodes. Our goal is to derive error estimates in the respective metrics, which allow for a validation of the model reduction approaches.

In a further step the application of Wasserstein-type metrics to more complex systems on networks without gradient structure shall be investigated, in particular compressible Euler-equations with friction, which are of central importance for modelling gas networks. The simple connection between Wasserstein-type metrics and Lagrangian formulations in one dimension shall be used to simplify the analysis on the one hand, but also to construct and analyze Lagrangian numerics in a systematic way.

A final step will investigate the application to optimization problems in network models. A particular goal is to derive optimality conditions for the state variable and the constraint, i.e. the partial differential equation, in Wasserstein-type space on networks. This yields a novel representation of the adjoint variable and allows for new approaches to the numerical optimization.
The derivation of mathematical methods in this projects is mainly driven by applications to gas networks, the rather general mathematical setup allows applications to other problems however, e.g. neuronal networks or biological transport networks, but also transport problems in higher spatial dimensions with non-homogeneous boundary and interface conditions.