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.

In this study, we investigate the numerical stability of the covariant BSSN (cBSSN) formulation proposed by Brown. We calculate the constraint amplification factor (CAF), which is an eigenvalue of the coefficient matrix of the evolution equations of the constraints on the cBSSN formulation and some adjusted formulations with constraints added to the evolution equations. The adjusted formulations have a higher numerical stability than the cBSSN formulation from the viewpoint of the CAF.

We perform simulations in a gravitational collapsing model using the Einstein equations. In this paper, we review the equations for constructing the initial values and the evolution form of the Einstein equations called the BSSN formulation. In addition, since we treat a nonvacuum case, we review the evolution equations of the matter fields of a perfect fluid. To make the simulations stable, we propose a modified system, which decreases numerical errors in analysis, and we actually perform stable simulations with decreased numerical errors.