Analyzing time evolution of constraint equations of Einstein's equation

Abstract

We propose a method of analyzing the time evolution of constraint equations of the Einstein's equation on the nonflat background by adding constraint terms to the evolution equations. In past studies, the eigenvalues of the coefficient matrix of the constraint equations was mostly obtained on the flat background. Since simulations do not always perform near the flat background, we need to calculate the eigenvalues on the appropriate background. We analyze eigenvalues numerically and predict the stability in the appropriate background. We also perform some numerical simulations and show consistency between the results of the eigenvalue analysis and the numerical stability.

On the stability of covariant BSSN formulation

“…In this paper, we suggested some adjustments and calculated the CAFs. There are several reported cases where the stability of the CAFs coincides with that of the numerical calculations [42,43,49,50]; thus, it is expected that the numerical calculations are more stable when using the adjustments. This is under investigation and will be reported in the near future.…”

Stable numerical simulation of Einstein equations in gravitational collapse space-time

“…Simply by adding terms consisting of constraints to the dynamical equations, stable and accurate simulations become possible. The details can be found in [6].…”

Parametrized numerical scheme for the Einstein equations

“…In these equations, the time and space are treated equivalently; and thus in performing numerical calculations, spacetime must be split into time and space (3+1 decomposition). One of the most famous formulations for numerical calculations of the Einstein equations is the Arnowitt-Desser-Misner (ADM) formulation [3][4][5], in which the Einstein equations are split into time evolution equations and constraint equations.…”

“…However, such common numerical schemes are not always appropriate for solving the Einstein equations, because the values of the constraints increase. So far, the numerical stability of the Einstein equations has been studied by considering the constraint amplification factor (CAF) [5,6]. Constraint-preserving schemes that enable the numerical calculations without the violation of constraints are also considered under certain conditions [7].…”