  A numerical study of the Einstein field equations, based on the 3+1 foliation
of the spacetime, is presented. A pseudo-spectral technique has been employed
for simulations in vacuum, within two different formalisms, namely the
Arnowitt-Deser-Misner (ADM) and the conformal
Baumgarte-Shapiro-Shibata-Nakamura (BSSN) approach. The numerical code is based
on the Fourier decomposition, accompanied by different filtering techniques.
The role of the dealiasing, as well as the influence of the filter type, has
been investigated. The algorithms have been stabilized via a novel procedure
that controls self-consistently the regularity of the solutions. The accuracy
of the model has been validated through standard testbeds, revealing that the
filtered pseudo-spectral technique is among the most accurate approaches.
Finally, the procedure has been stressed via black hole dynamics and a new
strategy, based on hyperviscous dissipation that suppresses spurious boundary
problems, has been proposed. The model represents a valid tool of
investigation, particularly suitable for the inspection of small scale
nonlinear phenomena in gravitational dynamics.
