Formulations of the Einstein equations for numerical simulations

We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations. In order to complete a long-term and accurate simulations of binary compact objects, people seek a robust set of equations against the violation of constraints. Many trials have revealed that mathematically equivalent sets of evolution equations show different numerical stability in free evolution schemes. In this article, we overview the efforts of the community, categorizing them into three directions: (1) modifications of the standard Arnowitt-Deser-Misner equations initiated by the Kyoto group (the so-called Baumgarte-Shapiro-Shibata-Nakamura equations), (2) rewriting the evolution equations in a hyperbolic form, and (3) construction of an "asymptotically constrained" system. We then introduce our series of works that tries to explain 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 damp the constraint-violating modes. We show several set of adjusted ADM/BSSN equations, together with their numerical demonstrations.