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We consider the numerical approximation of compressible flow in a pipenetwork. Appropriate coupling conditions are formulated that allow us to derivea variational characterization of solutions and to prove global balance lawsfor the conservation of mass and energy on the whole network. This variationalprinciple, which is the basis of our further investigations, is amenable to aconforming Galerkin approximation by mixed finite elements. The resultingsemi-discrete problems are well-posed and automatically inherit the globalconservation laws for mass and energy from the continuous level. We alsoconsider the subsequent discretization in time by a problem adapted implicittime stepping scheme which leads to conservation of mass and a slightdissipation of energy of the full discretization. The well-posedness of thefully discrete scheme is established and a fixed-point iteration is proposedfor the solution of the nonlinear systems arising in every single time step.Some computational results are presented for illustration of our theoreticalfindings and for demonstration of the robustness and accuracy of the newmethod.

A robust conservative mixed finite element method for compressible flow on pipe networks