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

对三维代码进行了改进，用于研究聚结的双中子星或黑洞和引力辐射。比较了几种具有不同形式的爱因斯坦方程的代码，其中包括调整后的ADM形式。为了将度规的演化方程化为双曲型偏微分方程组，人们提出了一些形式，但我们发现BSSN形式或它的修正在整体上对于数值精度和稳定性是非常好的。即使有些方法对于球对称或轴对称情况是有效的，但对于三维情况却不是这样。特别是对于较大的网格尺寸，更复杂的形式往往会导致数值不稳定，因为它会导致较大的数值误差。事实上，使用约束方程的形式主义可能会在时间演化过程中更好地保持约束，但它并不总是保持数值稳定性。此外，为保证数值稳定性，应尽可能精确地求解椭圆型偏微分方程组。由于求解椭圆型方程需要大量的CPU时间，因此需要一种高效的算法。将一些坐标条件化为椭圆型方程。然而，其中一些问题不需要精确地解决。相反，它们可以被近似地解决。在研究的后半部分，我们构造了一个合并黑洞的模拟代码。得到了一种具有奇异性剔除的稳定码。然而，我们使用的网格尺寸太小，不适用于更逼真的二进制块洞，需要进一步改进才能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.