Advantages of modified ADM formulation: constraint propagation analysis of Baumgarte-Shapiro-Shibata-Nakamura system

TL;DR

This paper analyzes the stability advantages of the BSSN formulation over the standard ADM in numerical relativity by eigenvalue analysis of constraint propagation, revealing how adjustments improve stability and proposing new modifications.

Contribution

It provides a detailed eigenvalue analysis of constraint propagation in the BSSN formulation, explaining its stability and suggesting potential improvements.

Findings

Adjustments in the BSSN equations enhance stability.

Constraint violations tend to decay with proper adjustments.

Proposed new adjustments could further improve stability.

Abstract

Several numerical relativity groups are using a modified ADM formulation for their simulations, which was developed by Nakamura et al (and widely cited as Baumgarte-Shapiro-Shibata-Nakamura system). This so-called BSSN formulation is shown to be more stable than the standard ADM formulation in many cases, and there have been many attempts to explain why this re-formulation has such an advantage. We try to explain the background mechanism of the BSSN equations by using eigenvalue analysis of constraint propagation equations. This analysis has been applied and has succeeded in explaining other systems in our series of works. We derive the full set of the constraint propagation equations, and study it in the flat background space-time. We carefully examine how the replacements and adjustments in the equations change the propagation structure of the constraints, i.e. whether violation of constraints (if it exists) will decay or propagate away. We conclude that the better stability of the BSSN system is obtained by their adjustments in the equations, and that the combination of the adjustments is in a good balance, i.e. a lack of their adjustments might fail to obtain the present stability. We further propose other adjustments to the equations, which may offer more stable features than the current BSSN equations.