  We describe a set of time evolution equations and its numerical
implementation for the investigation of non-axisymmetric oscillations of
rapidly rotating compact objects in full General Relativity, taking into
account the contribution of a dynamic spacetime. We derive the perturbation
equations for the spacetime in the Hilbert gauge, while the hydrodynamical
evolution is based on perturbations of the energy-momentum tensor. In our
numerical implementation, we use Kreiss-Oliger dissipation in order to achieve
a stable time evolution. Our code features high accuracy at comparably low
computational expense and we are able to extract the frequencies of
non-axisymmetric modes of compact objects with rotation rates up to the Kepler
limit.
