  We study a class of stochastic semilinear damped wave equations driven by
additive Wiener noise. Owing to the damping term, under appropriate conditions
on the nonlinearity, the solution admits a unique invariant distribution. We
apply semi-discrete and fully-discrete methods in order to approximate this
invariant distribution, using a spectral Galerkin method and an exponential
Euler integrator for spatial and temporal discretization respectively. We prove
that the considered numerical schemes also admit unique invariant
distributions, and we prove error estimates between the approximate and exact
invariant distributions, with identification of the orders of convergence. To
the best of our knowledge this is the first result in the literature concerning
numerical approximation of invariant distributions for stochastic damped wave
equations.
