The flow of compressible fluids through pipe networks is of big relevance, e.g., in the daily operation and capacity planning of gas transport networks. This talk gives an introduction to the governing equations and underlying physics and then develops a mathematical framework for the systematic discretization and numerical analysis of such problems.  

The talk will start with a short introduction to the basic equations of compressible fluid dynamics which are relevant for the flow of gas in long pipes. Based on the underlying conservation laws for mass, momentum, and energy, we will also formulate thermodynamically consistent coupling conditions for the pipe junctions.  

We then consider in detail isentropic flow in a single pipe and present an equivalent reformulation of the governing equations representing conservation of mass and the balance of kinetic energy. These new set of equations allows to establish a variational characterization of weak solutions which is very directly linked to conservation of mass and total energy.  

In a third step, we discuss the systematic discretization of the variational principle by Galerkin approximation in space via mixed finite elements and investigate the discretization in time by a problem adapted Runge-Kutta method. Due to the particular structure of the numerical approximation, we can prove exact conservation of mass and energy on the discrete level, up to numerical dissipation due to the implicit time stepping scheme. 

In the last part of the talk, we will show that the above variational principle and discretization strategy for one pipe can be generalized directly to pipe networks and then present some simulation results demonstrating the robustness of the method even in the presence of shocks. In addition, we will comment on the possible generalization to non-isentropic flow.