Formulation of Einstein equation for stable numerical simulation

爱因斯坦方程的数值模拟对于引力波和宇宙学的观测是必不可少的。但在真实的数值模拟中，特别是在强引力区或很长时间的模拟中，约束的违反往往会在时间演化中发散。为了准确和稳定的模拟，爱因斯坦方程的重新表述是必要的。这就是数值相对论的表述问题。本研究的目的是提出这一问题的标准的理论框架，并通过数值实验进行验证。作为补贴发放前的准备情况，我们建议采用约束放大因子(CAF)方法，并对约束传播进行特征值分析。可以说，这种方法在一定程度上是成功的，因为根据理论给出了大量的数值计算结果。然而，也存在许多问题。即使CAF很好，在很长一段时间内也会出现数值数据发散爬行的情况。此外，在约束保持的情况下，是否给出了正确的解还存在疑问。在此研究期间，我们可以提出许多调整后的配方，并进行CAF评估和数值验证。它被许多数值相对论研究者引用并产生了结果，此外，由于约束的传播类型避免了成为第二种约束，我们成功地更多地推迟了计算寿命。然而，计算寿命延长了，而且时间长了就发散了。因此，还需要进一步改进。此外，在已知精确解的情况下，我们通过数值模拟得到了闭合解。当然，这不能说是计算的正确证据，因为这只是一个例子。

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.