Comments on Bona-Masso type slicing conditions in long-term black hole evolutions

TL;DR

This paper analyzes Bona-Masso slicing conditions in black hole simulations, highlighting issues with gauge shocks and ill-posedness, and discusses conditions for reaching steady states and the implications for numerical methods.

Contribution

It characterizes steady states of Bona-Masso slicing, demonstrates their reachability in simulations, and identifies problems with current implementations, proposing potential solutions.

Findings

Steady states can be achieved with excision but not with the puncture method unless resolution is high.

BMn slicings often develop gauge shocks during evolution.

Current excision methods with BMn are ill-posed and need correction.

Abstract

We review in generality why time-independent endstates can be reached in black hole and collapse simulations, with and without excision. We characterise the Killing states of the Bona-Masso slicing condition with time derivative along the normals to the slice ("BMn") as solutions of a mixed elliptic/hyperbolic differential equation on the slice. We show numerically that these steady states can be reached as end states from typical initial data with excision but can be reached with the puncture method only if the puncture is not numerically well resolved. During the evolution, BMn slicings often form gauge shocks. It may be that these are not seen in current 3D simulations only through lack of resolution, although we expect that they can be avoided with some care. Finally we point out that excision with BMn as currently implemented is ill-posed and therefore not expected to converge; this can be cured. In technical appendixes, we derive the equations of pure gauge systems on a fixed spacetime, and bring the BSSN/NOR equations into 3-dimensional tensor form suitable for multiple coordinate patches or spherical polar coordinates.