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

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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.