Formulation of Einstein equation for stable numerical simulation

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Numerical simulation of Einstein equation is essential for observation of gravitational wave and cosmology. But in real numerical simulation, especially in strong gravitational region or in very long time simulation, the violation of constraint often diverges in the time evolution. For accurate and stable simulation, reformulations of the Einstein equations are necessary. This is the formulation problem of numerical relativity. The purpose of this research is to propose theoretical framework of this criterion of this problem and to demonstrate it by a numerical experiment. As the preparations situation before the grant of this subsidy, we suggests the Constraint Amplification Factor(CAF) approach with eigenvalue analysis of constraint propagation. It may be said that this approach is succeeded to some extent, because much numerical computation result according to the theory was given. However, there remain many problems, too. A lot of situation that the numerical data diverge crawled during long time even if CAF was good. In addition, there remains the doubt whether it gives right solution if constraint preserve. In this study period, we could present many adjusted formulations with evaluation by CAF and the numerical proof for it. It quoted by many researchers of numerical relativity and produces result In addition, we succeeded in postponing calculation life more because the propagation type of the constraint avoided becoming the second. However, the calculation life was prolonged, and it diverges after long time. So further improvement will be demanded. In addition, in some case that a exact solution was known, we see that a close solution is gotten by numerical simulation. Of course it cannot be said to right evidence for calculating because this is only one example.