We review recent efforts to re-formulate the Einstein equations for fullyrelativistic numerical simulations. The so-called numerical relativity(computational simulations in general relativity) is a promising research fieldmatching with ongoing astrophysical observations such as gravitational waveastronomy. Many trials for longterm stable and accurate simulations of binarycompact objects have revealed that mathematically equivalent sets of evolutionequations show different numerical stability in free evolution schemes. In thisarticle, we first review the efforts of the community, categorizing them intothe following three directions: (1) modifications of the standardArnowitt-Deser-Misner equations initiated by the Kyoto group, (2) rewriting ofthe evolution equations in hyperbolic form, and (3) construction of an"asymptotically constrained" system. We next introduce our idea for explainingthese evolution behaviors in a unified way using eigenvalue analysis of theconstraint propagation equations. The modifications of (or adjustments to) theevolution equations change the character of constraint propagation, and severalparticular adjustments using constraints are expected to diminish theconstraint-violating modes. We propose several new adjusted evolutionequations, and include some numerical demonstrations. We conclude by discussingsome directions for future research.
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