  We consider the simulation of barotropic flow of gas in long pipes and pipe
networks. Based on a Hamiltonian reformulation of the governing system, a fully
discrete approximation scheme is proposed using mixed finite elements in space
and an implicit Euler method in time. Assuming the existence of a smooth
subsonic solution bounded away from vacuum, a full convergence analysis is
presented based on relative energy estimates. Particular attention is paid to
establishing error bounds that are uniform in the friction parameter. As a
consequence, the method and results also cover the parabolic problem arising in
the asymptotic large friction limit. The error estimates are derived in detail
for a single pipe, but using appropriate coupling conditions and the particular
structure of the problem and its discretization, the main results directly
generalize to pipe networks. Numerical tests are presented for illustration.
