We consider the construction and analysis of observers for gas networks. The observer system is (nearly) a copy of the original system which, in our case, is governed by the barotropic Euler equations. The observer receives measurements of certain point values of the original system and uses them as boundary conditions. In this talk we will address two specific questions: 1) Suppose measurements are given at every boundary node of the network, how many internal measurement points are needed -- and where should they be located -- so that we can guarantee that the state of the observer system approximates the original system state for long times. 2) What happens if the observer system uses a different set of PDEs than the original system? We will show that if the two PDE systems are similar (in a sense that we will make precise) then the state of the observer will end up in a neighborhood of the original system state whose size is proportional to the difference between the two systems. This is joint work with M. Gugat (Erlangen) and T. Kunkel and V. Kumar (Darmstadt)

We study the dynamic formulation of the Schrödinger problem on metric graphs. Using the direct method of calculus of variations, we show existence of minimizers and investigate the connection to dynamic optimal transport. A particular focus lies on the analysis of $\Gamma$-convergence between both problems for vanishing diffusive effects. We extend our analytical findings by numerical examples based on an augmented lagrangian formulation and primal dual methods.

Over recent years, non-local conservation laws gained growing interest, especially for traffic flow. The well-known Lighthill-Whitham-Richards model has been extended by considering non-local velocity terms depending on the downstream traffic so that drivers adapt their velocity to the mean traffic in front. The motivation behind these kind of models lies in the progress of vehicular traffic and the growing interest in autonomous cars. These cars can be connected such that more than just one car in front of a driver is important for the decision of the driver's velocity. Therefore, the downstream traffic in a certain interaction range plays a significant role. In this talk we will consider a model in which drivers adapt their speed based on a mean downstream velocity. We will present well-posedness results based on a Godunov-type numerical scheme. Furthermore, we will investigate numerically what happens if the interaction range tends to zero and what happens if it tends to infinity (so every car is connected). Finally, we will extend the model to traffic flow networks, again providing numerical examples demonstrating the limt cases.

We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions in the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional. This talk is based on a joint work with G. M. Coclite, M. Garavello, and F. Marcellini.

The transition to hydrogen as an energy source poses various challenges and motivates the study of gas networks transporting gas mixtures. In this talk, we present a class of models for compressible binary gas mixture, with two velocity functions and both interior and pipe wall friction. For a broad class of pressure laws thermodynamic consistency of the model is guaranteed. Thanks to their formulation as port-Hamiltonian systems stability properties based on relative energy estimates can be proved using the general framework by [Egger, Giesselmann ‘23]. Furthermore, we present a numerical scheme that is asymptotic-preserving in the low Mach number and high-friction regime.

Projection-based model reduction is used to reduce the computational burden, e.g., for large-scale sub-systems of networks. When reducing nonlinear systems, the model order reduction step must be complemented by so-called complexity reduction methods to obtain a gain in efficiency. We focus on complexity reduction using quadrature/cubature-based method. This method has several advantages over the empirical interpolation-based approaches, including better online-performance as well as structure-preserving properties. However, the cost of the offline training is restrictively high in a large-scale setting. In this contribution, we propose a new training algorithm based on a constrained data compression approach that exploits the specific structure inherent in the training problem. Its efficiency is numerically demonstrated for network problems and compared to the standard approaches based on empirical interpolation. Furthermore, the structure-preserving properties of the empirical cubature/quadrature models are outlined.

Following ideas of van den Berg, Davies and Pólya, I am going to introduce the heat content of metric graphs and its integral version, the torsional rigidity. These objects have a similar and, to some extent, parallel theory. After describing how to obtain geometric bounds on them using graph surgical tools, I will discuss the possibility of delivering bounds on the ground state energy by means of the torsional rigidity. This is joint work with Marvin Plümer and Patrizio Bifulco.

Optimal transport is of great importance in gas networks and we utilize dynamic formulations of the p-Wasserstein metric, minimizing the kinetic energy necessary for moving gas through the network, for this task. We model our network as a metric graph and include a variety of constraints for ensuring the physical plausibility of the transport. For feasible optimal transport plans, our modelling combines different versions of Kirchhoff’s law at vertices with the isothermal Euler equations in pipes as well as time-dependent and time-independent boundary conditions. Moreover, we use the p-Wasserstein metric to derive gradient flow structures with the minimizing movement scheme, where for p=3 we again obtain the complete system of isothermal Euler equations.