Aliasing Instabilities in the Numerical Evolution of the Einstein Field
  Equations
  The Einstein field equations of gravitation are characterized by cross-scale,
high-order nonlinear terms, representing a challenge for numerical modeling. In
an exact spectral decomposition, high-order nonlinearities correspond to a
convolution that numerically might lead to aliasing instabilities. We present a
study of this problem, in vacuum conditions, based on the $3+1$
Baumgarte-Shibata-Shapiro-Nakamura (BSSN) formalism. We inspect the emergence
of numerical artifacts, in a variety of conditions, by using the
Spectral-FIltered Numerical Gravity codE (\texttt{SFINGE}) - a pseudo-spectral
algorithm, based on a classical (Cartesian) Fourier decomposition. By
monitoring the highest $k-$modes of the dynamical fields, we identify the
culprits of the aliasing and propose procedures that cure such instabilities,
based on the suppression of the aliased wavelengths. This simple algorithm,
together with appropriate treatment of the boundary conditions, can be applied
to a variety of gravitational problems, including those related to massive
objects dynamics.
