Black hole evolution by spectral methods

TL;DR

This paper demonstrates that pseudospectral collocation methods can stably evolve black hole spacetimes over long periods, overcoming limitations of traditional finite differencing techniques and simplifying black hole excision.

Contribution

It introduces a pseudospectral collocation approach for black hole evolution, showing improved stability and simplicity over finite-difference methods in a hyperbolic Einstein's equations framework.

Findings

PSC method evolves black holes indefinitely without constraints

Black hole excision is trivial with PSC in hyperbolic formulation

Method shows promise for extension to 3D simulations

Abstract

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation (PSC) method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein's equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast to finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the extension of this method to three spatial dimensions.