TY - INPR A1 - Strubberg, Lea A1 - Lutz, Annette A1 - Börner, Pascal A1 - Pfetsch, Marc A1 - Skutella, Martin A1 - Klimm, Max T1 - Valid Cuts for the Design of Potential-based Flow Networks N2 - The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program. Its non-convexity, due to the poten- tial flow, together with the binary variables, indicating the decision to build a connection, make these problems challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the bi- nary variables. Further, we show that this class of inequalities can be sep- arated in polynomial for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-bound al- gorithm. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks. KW - potential based flows KW - topology optimization KW - MINLP Y1 - 2025 ER - TY - INPR A1 - Breitkopf, Jannik A1 - Gugat, Martin A1 - Ulbrich, Stefan T1 - Existence and Optimal Boundary Control of Classical Solutions to Networks of Quasilinear Hyperbolic Systems of Balance Laws N2 - We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels. It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones. We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls. KW - networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control Y1 - 2025 ER - TY - INPR A1 - Giesselmann, Jan A1 - Ranocha, Hendrik T1 - Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions N2 - We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings. Y1 - 2025 ER - TY - INPR A1 - Breiten, Tobias A1 - Karsai, Attila A1 - Mehrmann, Volker A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Tscherpel, Tabea A1 - Hiller, Benjamin A1 - Morandin, Riccardo A1 - Tischendorf, Caren T1 - A Catalog of Gas Network Models: PDEs, Coupling Conditions, and Numerical Schemes N2 - This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems. We do not claim completeness and refer in many places to the existing literature. Y1 - N1 - This is an updated version of [P. Domschke, B. Hiller, J. Lang, V. Mehrmann, R. Morandin, and C. Tischendorf. Gas Network Modeling: An Overview. Preprint, TRR 154, 2021], available at: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/411 ER - TY - INPR A1 - Brunk, Aaron A1 - Giesselmann, Jan A1 - Tscherpel, Tabea T1 - A posteriori existence of strong solutions to the Navier-Stokes equations in 3D N2 - Global existence of strong solutions to the three-dimensional incompressible Navier--Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out blow-up on a certain time interval, using only numerical solutions. In this work we present such a result for the Navier--Stokes equations subject to periodic boundary conditions, which makes use of a version of the celebrated blow-up criterion in the critical space $L^\infty(L^3)$ by Iskauriaza, Serëgin and Shverak (2003). Our approach is based on a conditional stability estimate in $L^2$ and $L^3$. The a posteriori criterion that, if satisfied, verifies existence of strong solutions, involves only negative Sobolev norms of the residual. We apply the criterion to numerical approximations computed with mixed finite elements and an implicit Euler time discretisation. A posteriori error estimates allow us to derive a fully computable criterion without imposing any extra assumptions on the solution. While limited to short time intervals, with sufficient computational resources in principle the criterion might allow for a verification over longer time intervals than what can be achieved by theoretical means. KW - Navier-Stokes KW - blow-up KW - a posteriori estimates KW - critical space KW - reconstruction Y1 - ER - TY - JOUR A1 - Chaumet, Aidan A1 - Giesselmann, Jan T1 - Convergence Analysis of a Fully Discrete Observer for Data Assimilation of the Barotropic Euler Equations N2 - We study the convergence of a discrete Luenberger observer for the barotropic Euler equations in one dimension, for measurements of the velocity only. We use a mixed finite element method in space and implicit Euler integration in time. We use a modified relative energy technique to show an error bound comparing the discrete observer to the original system's solution. The bound is the sum of three parts: an exponentially decaying part, proportional to the difference in initial value, a part proportional to the grid sizes in space and time and a part that is proportional to the size of the measurement errors as well as the nudging parameter. The proportionality constants of the second and third parts are independent of time and grid sizes. To the best of our knowledge, this provides the first error estimate for a discrete observer for a quasilinear hyperbolic system, and implies uniform-in-time accuracy of the discrete observer for long-time simulations. KW - Data Assimilation KW - Observer KW - Relative Energy KW - Euler Equations KW - Fully Discrete Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2603.10962 ER - TY - JFULL A1 - Börner, Pascal A1 - Pfetsch, Marc E. A1 - Ulbrich, Stefan T1 - Mixing of Gases in Stationary Networks: Properties and Optimization N2 - This paper deals with the mixing of gases in stationary networks. We first derive a model and pressure law for mixing, for which there is empirical evidence for its accuracy. The model is based on the change of the speed of sound in gas mixtures. We then consider stationary gas networks. The existence result for solutions of single gas flow on networks is extended to mixtures. Further, we establish an easy to check criterion for uniqueness of solutions on the network. This results in a uniqueness proof of solutions on all networks with a mixture of natural gas and low hydrogen percentages. We then develop a solution algorithm that alternates between the solution of a single gas problem and update of the mixture ratios. In a computational study, different model variants and their impact on performance are compared. Moreover, the increased complexity of solving stationary gas transport problems with mixing is evaluated. KW - gas network optimization KW - gas mixing KW - MINLP KW - global optimization Y1 - ER -