Except as noted below, I generally follow the sign and notation conventions of Wald . I assume that all spacetimes are globally hyperbolic, and for event-horizon finding I further assume asymptotic flatness; in this latter context is future null infinity. I use the Penrose abstract-index notation, with summation over all repeated indices. 4-indices range over all spacetime coordinates , and 3-indices range over the spatial coordinates in a spacelike slice . The spacetime coordinates are thus .
Indices range over generic angular coordinates on or on a horizon surface. Note that these coordinates are conceptually distinct from the 3-dimensional spatial coordinates . Depending on the context, may or may not have the usual polar-spherical topology. Indices label angular grid points on or on a horizon surface. These are 2-dimensional indices: a single such index uniquely specifies an angular grid point. is the Kronecker delta on the space of these indices or, equivalently, on surface grid points.
For any indices and , and are the coordinate partial derivatives and respectively; for any coordinates and , and are the coordinate partial derivatives and respectively. is the flat-space angular Laplacian operator on , while refers to a finite-difference grid spacing in some variable .
is the spacetime 4-metric, and the inverse spacetime 4-metric; these are used to raise and lower 4-indices. are the 4-Christoffel symbols. is the Lie derivative along the 4-vector field .
I use the 3 + 1 “ADM” formalism first introduced by Arnowitt, Deser, and Misner ; York  gives a general overview of this formalism as it is used in numerical relativity. is the 3-metric defined in a slice, and is the inverse 3-metric; these are used to raise and lower 3-indices. is the associated 3-covariant derivative operator, and are the 3-Christoffel symbols. and are the 3 + 1 lapse function and shift vector respectively, so the spacetime line element is and York  in defining the extrinsic curvature of the slice as , where is the future-pointing unit normal to the slice. (In contrast, Wald  omits the minus signs from this definition.) is the trace of the extrinsic curvature . is the ADM mass of an asymptotically flat slice.
I often write a differential operator as , where the “” notation means that is a (generally nonlinear) algebraic function of the variable and its 1st and 2nd angular derivatives, and that also depends on the coefficients , , and at the apparent horizon position.
There are three common types of spacetimes/slices where numerical event or apparent horizon finding is of interest: spherically-symmetric spacetimes/slices, axisymmetric spacetimes/slices, and spacetimes/slices with no continuous spatial symmetries (no spacelike Killing vectors). I refer to the latter as “fully generic” spacetimes/slices.
In this review I use the abbreviations “ODE” for ordinary differential equation, “PDE” for partial differential equation, “CE surface” for constant-expansion surface, and “MOTS” for marginally outer trapped surface. Names in Small Capitals refer to horizon finders and other computer software.
When discussing iterative numerical algorithms, it is often convenient to use the concept of an algorithm’s “radius of convergence”. Suppose the solution space within which the algorithm is iterating is . Then given some norm on , the algorithm’s radius of convergence about a solution is defined as the smallest such that the algorithm will converge to the correct solution for any initial guess with . We only rarely know the exact radius of convergence of an algorithm, but practical experience often provides a rough estimate1.
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