Constraint Propagation of $C^2$-adjusted Formulation II --- Another Recipe for Robust Baumgarte-Shapiro-Shibata-Nakamura Evolution System ---

TL;DR

This paper introduces a new $C^2$-adjusted BSSN evolution system that enhances robustness against constraint violations, significantly extending simulation lifetimes in numerical relativity tests.

Contribution

It develops a $C^2$-adjusted formulation for BSSN equations, including all constraints, and demonstrates improved stability and longer simulation durations.

Findings

Simulation lifetime increased by 2-10 times.

Including all constraints in $C^2$ improves stability.

Numerical tests confirm effectiveness in gauge-wave and Gowdy spacetimes.

Abstract

In order to obtain an evolution system which is robust against the violation of constraints, we present a new set of evolution systems based on the so-called Baumgarte-Shapiro-Shibata-Nakamura (BSSN) equations.The idea is to add functional derivatives of the norm of constraints, $C^2$, to the evolution equations, which was proposed by Fiske (2004) and was applied to the ADM formulation in our previous study. We derive the constraint propagation equations, discuss the behavior of constraint damping, and present the results of numerical tests using the gauge-wave and polarized Gowdy wave spacetimes. The construction of the $C^2$-adjusted system is straightforward. However, in BSSN, there are two kinetic constraints and three algebraic constraints; thus, the definition of $C^2$ is a matter of concern. By analyzing constraint propagation equations, we conclude that $C^2$ should include all the constraints, which is also confirmed numerically. By tuning the parameters, the lifetime of the simulations can be increased as 2-10 times as longer than those of the standard BSSN evolutions.