  We consider the numerical approximation of compressible flow in a pipe
network. Appropriate coupling conditions are formulated that allow us to derive
a variational characterization of solutions and to prove global balance laws
for the conservation of mass and energy on the whole network. This variational
principle, which is the basis of our further investigations, is amenable to a
conforming Galerkin approximation by mixed finite elements. The resulting
semi-discrete problems are well-posed and automatically inherit the global
conservation laws for mass and energy from the continuous level. We also
consider the subsequent discretization in time by a problem adapted implicit
time stepping scheme which leads to conservation of mass and a slight
dissipation of energy of the full discretization. The well-posedness of the
fully discrete scheme is established and a fixed-point iteration is proposed
for the solution of the nonlinear systems arising in every single time step.
Some computational results are presented for illustration of our theoretical
findings and for demonstration of the robustness and accuracy of the new
method.
