Will hyperbolic formulations help numerical relativity? - Experiments using Ashtekar's connection variables

TL;DR

This paper compares different hyperbolic formulations of Einstein's equations using Ashtekar's connection variables to evaluate their stability and accuracy in numerical relativity simulations.

Contribution

It introduces a unified approach to hyperbolic systems with Ashtekar variables and assesses asymptotically constrained systems for improved numerical stability.

Findings

Symmetric hyperbolic systems show enhanced stability.

Asymptotically constrained systems improve robustness against errors.

Using Ashtekar's variables allows consistent treatment across hyperbolicity levels.

Abstract

In order to perform accurate and stable long-term numerical integration of the Einstein equations, several hyperbolic systems have been proposed. We here report our numerical comparisons between weakly hyperbolic, strongly hyperbolic, and symmetric hyperbolic systems based on Ashtekar's connection variables. The primary advantage for using this connection formulation is that we can keep using the same dynamical variables for all levels of hyperbolicity. We also study asymptotically constrained systems, "$\lambda$-system" and "adjusted system", for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. These systems are tested in the Maxwell system and in the Ashtekar's system. This mechanism affects more than the system's symmetric hyperbolicity. (This workshop contribution is the summary of our gr-qc/0005003 [CQG 17 (2000) 4799] and gr-qc/0007034 [CQG 18 (2001) 441].)