  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.
