Galerkin methods for simulation of wave propagation on a network of pipes

In this work we study a class of semi-discrete hyperbolic partial differential equations on networks, which model the propagation of pressure waves in pipelines. We show well-posedness of the corresponding problem and derive exponential convergence to the solution of a stationary problem for time independent data. To discretize the problem, we suggest a Galerkin semi-discretization in space and a parametrized class of implicit Runge-Kutta schemes in time. Then the corresponding semi-discrete and fully-discrete problems are also shown to be well-posed, and the exponential stability estimates are proven to carry over to the discrete setting uniformly in the discretization parameters. Abstract discretization error estimates are shown, which provide explicit convergence rates for the choice of a mixed finite element realization of the Galerkin scheme. The theoretical findings are further compared and discussed to improved results that can be obtained for a linear approximation of the physical model. We illustrate our theoretical findings by numerical experiments. Furthermore, model reduction approaches based on Galerkin projections are studied, with the focus on how they can be modified so that the reduced model inherits certain properties from the physical model. As a result, this yields a high precision structure preserving low dimensional model, preserving the port-Hamiltonian structure, the conservation of the total mass, existence of steady states, and exponential convergence to those states for time-independent data. A numerical realization in finite precision arithmetics is discussed, and the necessity of the modification step is shown by numerical tests. Finally, we consider an optimal control problem on a network together with a realistic gas model. There, a compressor is controlled in an optimal sense, so that the demand at an exit point of the network is satisfied. This leads to an optimal control problem with control and state constraints, where the latter are included into the objective function by a barrier function. Existence of a minimizer is shown, and we utilize the Gauss-Newton method to find a discrete solution. The method is realized by solving adjoint problems for efficiency, and the scenario is solved numerically.