  We consider the discretization and subsequent model reduction of a system of
partial differential-algebraic equations describing the propagation of pressure
waves in a pipeline network. Important properties like conservation of mass,
dissipation of energy, passivity, existence of steady states, and exponential
stability can be preserved by an appropriate semi-discretization in space via a
mixed finite element method and also during the further dimension reduction by
structure preserving Galerkin projection which is the main focus of this paper.
Krylov subspace methods are employed for the construciton of the reduced models
and we discuss modifications needed to satisfy certain algebraic compatibility
conditions; these are required to ensure the well-posedness of the reduced
models and the preservation of the key properties. Our analysis is based on the
underlying infinite dimensional problem and its Galerkin approximations. The
proposed algorithms therefore have a direct interpretation in function spaces;
in principle, they are even applicable directly to the original system of
partial differential-algebraic equations while the intermediate discretization
by finite elements is only required for the actual computations. The
performance of the proposed methods is illustrated with numerical tests and the
necessity for the compatibility conditions is demonstrated by examples.
