Structure-Preserving Numerical Approximations for a Port-Hamiltonian Formulation of the Non-Isothermal Euler Equations

In this thesis we introduce infinite dimensional port-Hamiltonian formulations of a model library based on the compressible non-isothermal Euler equations to model pipe flow with temperature-dependence. Additionally, we set up the underlying Stokes-Dirac structures and deduce the boundary port variables. Following that, we adapt the structure-preserving semi-discretization for the isothermal Euler equations to the non-isothermal case. As these systems are highly non-linear we use the extended group finite element method to make the non-linearities easily manageable during model order and complexity reduction. These two procedures are necessary when simulating large networks of pipes in reasonable amounts of time. Thus, we deduce a structure-preserving model order reduction procedure for the single pipe system. Furthermore, we compare two complexity reduction procedures, i.e., the discrete empirical interpolation method and an empirical quadrature based ansatz, which is even structure-preserving. Finally, we introduce coupling conditions into our port-Hamiltonian formulations, such that the structure of the single pipes is preserved and the whole network system is port-Hamiltonian itself. As the port-Hamiltonian structure is preserved during coupling the numerical methods developed for the single pipe systems can be easily applied to the network case. Academic numerical examples will support our analytical findings.