Stability preserving approximations of a semilinear hyperbolic gas transport model

We consider the discretization of a semilinear damped wave equation arising, for instance, in the modeling of gas transport in pipeline networks. For time invariant boundary data, the solutions of the problem are shown to converge exponentially fast to steady states. We further prove that this decay behavior is inherited uniformly by a class of Galerkin approximations, including finite element, spectral, and structure preserving model reduction methods. These theoretical findings are illustrated by numerical tests.