  We numerically investigate the propagation of plane gravitational waves in
the form of an initial boundary value problem with de Sitter initial data. The
full non-linear Einstein equations with positive cosmological constant
$\lambda$ are written in the Friedrich-Nagy gauge which yields a wellposed
system. The propagation of a single wave and the collision of two with colinear
polarization are studied and contrasted with their Minkowskian analogues.
Unlike with $\lambda=0$, critical behaviours are found with $\lambda>0$ and are
based on the relationship between the wave profile and $\lambda$. We find that
choosing boundary data close to one of these bifurcations results in a "false"
steady state which violates the constraints. Simulations containing
(approximate) impulsive wave profiles are run and general features are
discussed. Analytic results of Woodard and Tsamis, which describe how
gravitational waves could affect an expansion rate at an initial instance of
time, are explored and generalized to the entire space-time. Finally we put
forward boundary conditions that, at least locally, slow down the expansion
considerably for a time.
