Three Dimensional Codes for Numerical Relativity with New Formalisms of the Einstein Equation

对三维程序进行了改进，用于研究并合双星、黑洞和引力辐射。几个代码与各种形式主义的爱因斯坦方程，其中包括调整ADM形式主义，进行了比较。一些形式化的度量的发展方程减少到双曲型偏微分方程的约束方程，但我们发现，BSSN形式或其修改作为一个整体是非常好的数值精度和稳定性。即使有些是有效的球或轴对称的情况下，他们是不正确的三维情况。特别是在大的网格尺寸，更复杂的形式往往会导致数值不稳定，因为它会导致更大的数值误差。事实上，使用约束方程的形式主义可以在时间演化过程中更好地保持约束，但它并不总是租赁数值稳定性。此外，椭圆型偏微分方程应尽可能精确地求解，以保证数值稳定性。由于求解椭圆方程需要大量的CPU时间，因此我们需要一种有效的算法。一些坐标条件被简化为椭圆方程。然而，其中一些不需要精确地解决。相反，它们可能会被近似求解。在研究的后半部分，我们构建了一个合并黑洞的模拟代码。得到了具有奇异性切除的稳定码。然而，我们使用的网格尺寸太小，适用于更现实的二进制块孔和进一步的改进，需要完全使用并行计算机。

Improvement of three dimensional code has been made to investigate coalescing binary neutron stars or black holes and gravitation radiation. Several codes with various formalism of the Einstein equation, which includes adjusted ADM formalism, has been compared. Some formalisms are proposed to reduce evolution equations of the metric into hyperbolic partial differential equations by means of constraints equations, but we found that the BSSN formalism or its modification is as a whole very good for numerical precision and stability. Even if some are efficient for spherically or axially symmetric cases, they are not true for three dimensional cases. In particular with a large grid size, more complicated formalism tends to leads numerical instability since it causes larger numerical errors. Indeed, a formalism using constraint equations may keep the constrain better in the course of time evolution, but it does not always lease numerical stability. In addition, elliptic partial differential equations should be solved as precisely as possible for numerical stability. Since it requires a lot of CPU hours to solve elliptic equations, we need an efficient algorism for them. Some coordinate conditions are reduced to Elliptic equations. However, some of them need not be solved precisely. Instead, they may be solved approximately.In the second half of the research, we construct a simulation code for coalescing black holes. A stable code with a singularity excision was obtained. However, the grid size we used is too small to apply to more realistic binary block holes and further improvement is required to use parallel computers entirely.