Aliasing Instabilities in the Numerical Evolution of the Einstein Field Equations

TL;DR

This paper investigates aliasing instabilities in spectral numerical simulations of Einstein's equations, identifies their causes, and proposes filtering techniques to suppress these artifacts, improving the stability of gravitational modeling.

Contribution

It introduces a spectral filtering method to mitigate aliasing instabilities in the numerical evolution of Einstein's equations using the BSSN formalism.

Findings

Aliasing causes numerical artifacts in spectral simulations of Einstein's equations.

Filtering high-frequency modes reduces aliasing instabilities.

The method improves stability in gravitational simulations.

Abstract

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.