Numerical stability of a new conformal-traceless 3+1 formulation of the Einstein equation

TL;DR

This paper introduces a new conformal-traceless formulation of the Einstein equations that improves the stability and lifetime of numerical black hole simulations compared to existing methods.

Contribution

A novel conformal-traceless system for Einstein equations with different transformations and non-linear term handling, enhancing numerical stability.

Findings

Extended simulation lifetime for black hole evolutions.

Demonstrated improved numerical stability over existing systems.

Validated the new formulation with 3D black hole simulations.

Abstract

There is strong evidence indicating that the particular form used to recast the Einstein equation as a 3+1 set of evolution equations has a fundamental impact on the stability properties of numerical evolutions involving black holes and/or neutron stars. Presently, the longest lived evolutions have been obtained using a parametrized hyperbolic system developed by Kidder, Scheel and Teukolsky or a conformal-traceless system introduced by Baumgarte, Shapiro, Shibata and Nakamura. We present a new conformal-traceless system. While this new system has some elements in common with the Baumgarte-Shapiro-Shibata-Nakamura system, it differs in both the type of conformal transformations and how the non-linear terms involving the extrinsic curvature are handled. We show results from 3D numerical evolutions of a single, non-rotating black hole in which we demonstrate that this new system yields a significant improvement in the life-time of the simulations.