A spectral method algorithm for numerical simulations of gravitational fields

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.

PBH Formation from Spherically Symmetric Hydrodynamical Perturbations: A Review

“…Pseudospectral methods have been extensively used in computer science for solving complex physics phenomena [159][160][161]. The application to the field of numerical relativity has been also very popular to study different problems, from astrophysics to more oriented toward high energy physics [162][163][164][165][165][166][167][168][169][170][171][172][173][174][175][176].…”

PBH formation from spherically symmetric hydrodynamical perturbations: a review

“…Pseudospectral methods have been extensively used in computer science for solving complex physics phenomena [156][157][158]. The application to the field of numerical relativity has been also very popular to study different problems, from astrophysics to more oriented on high energy physics [159][160][161][162][163][164][165][166][167][168][169][170][171][172][173][174].…”

Strong Interactions in the Three Black Holes Problem

“…The numerical procedure is described in [42] and has been tested against all classical testbeds. The spatial derivatives are computed in Cartesian geometry, via standard FFTs.…”

“…For the time integration we adopt a second-order Runge-Kutta method, with a time-step can be changed during the evolution, as described in [42]. Finally, we introduce an implicit, viscous strategy to absorb boundary disturbances, typical of the numerical tests that have a small violation of the periodicity.…”