Stability of the puncture method with a generalized Baumgarte-Shapiro- Shibata-Nakamura formulation

The puncture method for dealing with black holes in the numerical simulation of vacuum spacetimes is remarkably successful when combined with the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein equations. We examine a generalized class of formulations modeled along the lines of the Laguna-Shoemaker system and including BSSN as a special case. The formulation is a two parameter generalization of the choice of variables used in standard BSSN evolutions. Numerical stability of the standard finite difference methods is proven for the formulation in the linear regime around flat space, a special case of which is the numerical stability of BSSN. Numerical evolutions are presented and compared with a standard BSSN implementation. Surprisingly, a significant portion of the parameter space yields (long-term) stable simulations, including the standard BSSN formulation as a special case. Furthermore, nonstandard parameter choices typically result in smoother behavior of the evolution variables close to the puncture.