Improved fast-rotating black hole evolution simulations with modified Baumgarte-Shapiro-Shibata-Nakamura formulation

TL;DR

This paper introduces modifications to the BSSN formulation in numerical relativity that enhance stability and accuracy in simulating highly spinning black holes, significantly reducing unphysical angular momentum loss.

Contribution

The authors propose and experimentally validate three modifications to the BSSN formulation that improve stability, hyperbolicity, and dissipation control in black hole simulations.

Findings

Enhanced stability and accuracy in black hole evolutions.

Significant reduction in angular momentum loss.

Effective suppression of unphysical dissipation.

Abstract

Different formulations of Einstein's equations used in numerical relativity can affect not only the stability but also the accuracy of numerical simulations. In the original Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation, the loss of the angular momentum, $J$, is non-negligible in highly spinning single black hole evolutions. This loss also appears, usually right after the merger, in highly spinning binary black hole simulations, The loss of $J$ may be attributed to some unclear numerical dissipation. Reducing unphysical dissipation is expected to result in more stable and accurate evolutions. In the previous work \cite{yhlc12} we proposed several modifications which are able to prevent black hole evolutions from the unphysical dissipation, and the resulting simulations are more stable than in the traditional BSSN formulation. Specifically, these three modifications (M1, M2, and M3) enhance the effects of stability, hyperbolicity, and dissipation of the formulation. We experiment further in this work with these modifications, and demonstrate that these modifications improve the accuracy and also effectively suppress the loss of $J$, particularly in the black hole simulations with the initially large ratio of $J$ and the square of the ADM mass.