  We review recent efforts to re-formulate the Einstein equations for fully
relativistic numerical simulations. The so-called numerical relativity
(computational simulations in general relativity) is a promising research field
matching with ongoing astrophysical observations such as gravitational wave
astronomy. Many trials for longterm stable and accurate simulations of binary
compact objects have revealed that mathematically equivalent sets of evolution
equations show different numerical stability in free evolution schemes. In this
article, we first review the efforts of the community, categorizing them into
the following three directions: (1) modifications of the standard
Arnowitt-Deser-Misner equations initiated by the Kyoto group, (2) rewriting of
the evolution equations in hyperbolic form, and (3) construction of an
"asymptotically constrained" system. We next introduce our idea for explaining
these evolution behaviors in a unified way using eigenvalue analysis of the
constraint propagation equations. The modifications of (or adjustments to) the
evolution equations change the character of constraint propagation, and several
particular adjustments using constraints are expected to diminish the
constraint-violating modes. We propose several new adjusted evolution
equations, and include some numerical demonstrations. We conclude by discussing
some directions for future research.
