Spectral Methods for Numerical Relativity

TL;DR

This paper reviews spectral methods for solving complex equations in General Relativity, highlighting their fast convergence and application to various astrophysical phenomena, including black hole mergers and supernovae.

Contribution

It provides a comprehensive overview of spectral methods, including theoretical foundations, stability analysis, and their application to diverse problems in numerical relativity.

Findings

Spectral methods exhibit rapid convergence in solving PDEs in relativity.

Applications include initial data for black holes and simulations of astrophysical events.

Spectral evolution techniques are stable and effective for complex systems.

Abstract

Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers.