  We present a numerical study of the Einstein equations, according to the
Arnowitt-Deser-Misner (ADM) formalism, in order to simulate the dynamics of
gravitational fields. We took in consideration the original $3+1$ decomposition
of the ADM equations, in vacuum conditions, in simplified geometries. The
numerical code is based on spectral methods, making use of filtering
(de-aliasing) techniques. The algorithm has been stabilized via an adaptive
time-refinement, based on a procedure that checks self-consistently the
regularity of the solutions. The accuracy of our numerical model has been
validated through a series of standard tests. Finally, we present also a new
kind of initial data that can be used for testing numerical codes.
