Event and apparent horizon finders for 3+1 numerical relativity

Event and apparent horizons are key diagnostics for the presence and properties of black holes. In this article I review numerical algorithms and codes for finding event and apparent horizons in numerically-computed spacetimes, focusing on calculations done using the 3+1 ADM formalism. There are 3 basic algorithms for finding event horizons, based respectively on integrating null geodesics \emph{forwards} in time, integrating null geodesics \emph{backwards} in time, and integrating null \emph{surfaces} backwards in time. The last of these is generally the most efficient and accurate. There are a large number of apparent-horizon finding algorithms, with differing trade-offs between speed, robustness, accuracy, and ease of programming. In axisymmetry, shooting algorithms work well and are fairly easy to program. In slices with no continuous symmetries, Nakamura et al.'s algorithm and elliptic-PDE algorithms are fast and accurate, but require good initial guesses to converge. In many cases Schnetter's "pretracking" algorithm can greatly improve an elliptic-PDE algorithm's robustness. Flow algorithms are generally quite slow, but can be very robust in their convergence.