Conformal and covariant formulation of the Z4 system with constraint-violation damping

TL;DR

This paper introduces a new conformal and covariant formulation of the Z4 system for Einstein equations, combining advantages of existing methods to improve constraint violation damping and robustness in numerical relativity simulations.

Contribution

A novel conformal, covariant Z4 formulation that enhances constraint damping and is easily integrated into existing BSSNOK codes.

Findings

Better constraint behavior observed in tests

Rapid suppression of violations during evolutions

Minimal implementation changes and computational costs

Abstract

We present a new formulation of the Einstein equations based on a conformal and traceless decomposition of the covariant form of the Z4 system. This formulation combines the advantages of a conformal decomposition, such as the one used in the BSSNOK formulation (i.e. well-tested hyperbolic gauges, no need for excision, robustness to imperfect boundary conditions) with the advantages of a constraint-damped formulation, such as the generalized harmonic one (i.e. exponential decay of constraint violations when these are produced). We validate the new set of equations through standard tests and by evolving binary black hole systems. Overall, the new conformal formulation leads to a better behavior of the constraint equations and a rapid suppression of the violations when they occur. The changes necessary to implement the new conformal formulation in standard BSSNOK codes are very small as are the additional computational costs.