Abstract A numerical study of the Einstein field equations, based on 3 + 1 foliation spacetime, is presented. pseudo-spectral technique has been employed for simulations in vacuum, within two different formalisms, namely Arnowitt–Deser–Misner (ADM) and conformal Baumgarte–Shapiro–Shibata–Nakamura (BSSN) approach. The code Fourier decomposition, accompanied by filtering techniques. role dealiasi...