  In this paper we study a numerical implementation for the initial boundary
value formulation for the generalized conformal field equations. We propose a
formulation which is well suited for the study of the long-time behaviour of
perturbed exact solutions such as a Schwarzschild or even a Kerr black hole. We
describe the derivation of the implemented equations which we give in terms of
the space-spinor formalism. We discuss the conformal Gauss gauge, and a slight
generalization thereof which seems to be particularly useful in the presence of
boundaries. We discuss the structure of the equations at the boundary and
propose a method for imposing boundary conditions which allow the correct
number of degrees of freedom to be freely specified while still preserving the
constraints. We show that this implementation yields a numerically well-posed
system by testing it on a simple case of gravitational perturbations of
Minkowski space-time and subsequently with gravitational perturbations of
Schwarzschild space-time.
