A cure for unstable numerical evolutions of single black holes: adjusting the standard ADM equations

TL;DR

This paper demonstrates that adding constraint terms to the standard ADM equations can stabilize long-term numerical evolutions of single black holes, especially in spherical symmetry, by exploring the effects of gauge choices and introducing quasi well-posedness.

Contribution

It introduces a method of adjusting the ADM equations with constraint terms to improve stability in black hole simulations, and defines quasi well-posedness for ADM-like systems.

Findings

Adding constraint terms enhances stability of black hole evolutions.

Gauge choices significantly influence the stability of numerical simulations.

The concept of quasi well-posedness aids in analyzing ADM system stability.

Abstract

Numerical codes based on a direct implementation of the standard ADM formulation of Einstein's equations have generally failed to provide long-term stable and convergent evolutions of black hole spacetimes when excision is used to remove the singularities. We show that, for the case of a single black hole in spherical symmetry, it is possible to circumvent these problems by adding to the evolution equations terms involving the constraints, thus adjusting the standard ADM system. We investigate the effect that the choice of the lapse and shift has on the stability properties of numerical simulations and thus on the form of the added constraint term. To facilitate this task, we introduce the concept of quasi well-posedness, a version of well-posedness suitable for ADM-like systems involving second-order spatial derivatives.