Constraint damping for the Z4c formulation of general relativity

TL;DR

This paper investigates the effectiveness of a constraint damping scheme in numerical relativity, analyzing its impact on various spacetime simulations and emphasizing the importance of parameter selection for stable, accurate evolutions.

Contribution

It extends the analysis of the constraint damping scheme to numerical applications involving conformal Z4, providing practical insights into its benefits and limitations.

Findings

Damping scheme effectively reduces constraint violations when properly resolved on the grid.

Artificial dissipation combined with damping suppresses constraint violations on grid noise.

Improper damping parameter choices can lead to unphysical evolutions and constraint growth.

Abstract

One possibility for avoiding constraint violation in numerical relativity simulations adopting free-evolution schemes is to modify the continuum evolution equations so that constraint violations are damped away. Gundlach et. al. demonstrated that such a scheme damps low amplitude, high frequency constraint violating modes exponentially for the Z4 formulation of General Relativity. Here we analyze the effect of the damping scheme in numerical applications on a conformal decomposition of Z4. After reproducing the theoretically predicted damping rates of constraint violations in the linear regime, we explore numerical solutions not covered by the theoretical analysis. In particular we examine the effect of the damping scheme on low-frequency and on high-amplitude perturbations of flat spacetime as well and on the long-term dynamics of puncture and compact star initial data in the context of spherical symmetry. We find that the damping scheme is effective provided that the constraint violation is resolved on the numerical grid. On grid noise the combination of artificial dissipation and damping helps to suppress constraint violations. We find that care must be taken in choosing the damping parameter in simulations of puncture black holes. Otherwise the damping scheme can cause undesirable growth of the constraints, and even qualitatively incorrect evolutions. In the numerical evolution of a compact static star we find that the choice of the damping parameter is even more delicate, but may lead to a small decrease of constraint violation. For a large range of values it results in unphysical behavior.