A spectral method algorithm for numerical simulations of gravitational fields

TL;DR

This paper presents a highly accurate spectral method for simulating gravitational fields using Einstein's equations, employing Fourier-based filtering and stabilization techniques, and demonstrates its effectiveness in black hole dynamics.

Contribution

Introduces a novel spectral algorithm with filtering and stabilization for Einstein equations, improving accuracy and handling small-scale nonlinear gravitational phenomena.

Findings

Validated with standard testbeds showing high accuracy

Effective stabilization via a new regularity control procedure

Successful simulation of black hole dynamics with reduced boundary issues

Abstract

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.