Aliasing Instabilities in the Numerical Evolution of the Einstein Field Equations

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.

References in corpus (17)

• Magnetic field generation in fully convective rotating spheres
• Binary black hole merger dynamics and waveforms
• Solving Einstein's Equations With Dual Coordinate Frames
• Hyperviscosity, Galerkin truncation and bottlenecks in turbulence
• A waveform model for eccentric binary black hole based on effective-one-body-numerical-relativity (EOBNR) formalism
• Simulation of Binary Black Hole Spacetimes with a Harmonic Evolution Scheme
• A lower bound on the maximum mass if the secondary in GW190814 was once a rapidly spinning neutron star
• Covariant formulations of BSSN and the standard gauge
• Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions
• Foundations of multiple black hole evolutions
• Modeling the source of GW150914 with targeted numerical-relativity simulations
• Implementation of standard testbeds for numerical relativity
• Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity
• A Reinvestigation of Moving Punctured Black Holes with a New Code
• BSSN in Spherical Symmetry
• Stability of the puncture method with a generalized BSSN formulation
• Constraint preserving boundary conditions for the linearized BSSN formulation