  Motivated by the problem of the evolution of bulk gravitational waves in
Randall-Sundrum cosmology, we develop a characteristic numerical scheme to
solve 1+1 dimensional wave equations in the presence of a moving timelike
boundary. The scheme exhibits quadratic convergence, is capable of handling
arbitrary brane trajectories, and is easily extendible to non-AdS bulk
geometries. We use our method to contrast two different prescriptions for the
bulk fluctuation initial conditions found in the literature; namely, those of
Hiramatsu et al. (hep-th/0410247) and Ichiki and Nakamura (astro-ph/0406606).
We find that if the initial data surface is set far enough in the past, the
late time waveform on the brane is insensitive to the choice between the two
possibilities; and we present numeric and analytic evidence that this
phenomenon generalizes to more generic initial data. Observationally, the main
consequence of this work is to re-affirm previous claims that the stochastic
gravitational wave spectrum is predominantly flat, in contradiction with naive
predictions from the effective 4-dimensional theory. Furthermore, this flat
spectrum result is predicted to be robust against uncertainties in (or
modifications of) the bulk initial data, provided that the energy scale of
brane inflation is high enough.
