Numerical Relativity with Arbitrary Precision Arithmetic: Applications to Gravitational Collapse

TL;DR

This paper demonstrates the use of high-precision pseudo-spectral methods in numerical relativity, enabling highly accurate simulations of gravitational collapse phenomena such as Choptuik collapse and AdS spacetime collapse.

Contribution

It introduces a novel combination of PSC methods with arbitrary precision arithmetic for improved accuracy and efficiency in numerical relativity simulations.

Findings

High-precision PSC methods accurately locate apparent horizons.

Numerical evolution preserves total energy in AdS collapse with high precision.

Exponential convergence achieved for smooth relativistic problems.

Abstract

Numerical Relativity is a mature field with many applications in Astrophysics, Cosmology and even in Fundamental Physics. As such, we are entering a stage in which new sophisticated methods adapted to open problems are being developed. In this paper, we advocate the use of Pseudo-Spectral Collocation (PSC) methods in combination with high-order precision arithmetic for Numerical Relativity problems with high accuracy and performance requirements. The PSC method provides exponential convergence (for smooth problems, as is the case in many problems in Numerical Relativity) and we can use different bit precision without the need of changing the structure of the numerical algorithms. Moreover, the PSC method provides high-compression storage of the information. We introduce a series of techniques for combining these tools and show their potential in two problems in relativistic gravitational collapse: (i) The classical Choptuik collapse, estimating with arbitrary precision the location of the apparent horizon. (ii) Collapse in asympotically anti-de Sitter spacetimes, showing that the total energy is preserved by the numerical evolution to a very high degree of precision.