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Part I
Introduction

Systems with strong gravitational fields, particularly systems which may contain event horizons and/or apparent horizons, are a major focus of numerical relativity. The usual output of a numerical relativity simulation is some (approximate, discrete) representation of the spacetime geometry (the 4-metric and possibly its derivatives) and any matter fields, but not any explicit information about the existence, precise location, or other properties of any event/apparent horizons. To gain this information, we must explicitly find the horizons from the numerically-computed spacetime geometry. The subject of this review is numerical algorithms and codes for doing this, focusing on calculations done using the 3 + 1 ADM formalism [14Jump To The Next Citation Point163Jump To The Next Citation Point]. Baumgarte and Shapiro [27Jump To The Next Citation Point, Section 6] have also recently reviewed event and apparent horizon finding algorithms. The scope of this review is limited to the finding of event/apparent horizons and omits any but the briefest mention of the many uses of this information in gaining physical understanding of numerically-computed spacetimes. In this review I distinguish between a numerical algorithm (an abstract description of a mathematical computation; also often known as a “method” or “scheme”), and a computer code (a “horizon finder”, a specific piece of computer software which implements a horizon finding algorithm or algorithms). My main focus is on the algorithms, but I also mention specific codes where they are freely available to other researchers. In this review I have tried to cover all the major horizon finding algorithms and codes, and to accurately credit the earliest publication of important ideas. However, in a field as large and active as numerical relativity, it is not unlikely that I have overlooked and/or misdescribed some important research. I apologise to anyone whose work I’ve slighted, and I ask readers to help make this a truly “living” review by sending me corrections, updates, and/or pointers to additional work (either their own or others) that I should discuss in future revisions of this review. The general outline of this review is as follows: In the remainder of Part I, I define notation and terminology (Section 1), discuss how 2-surfaces should be parameterized (Section 2), and outline some of the software-engineering issues that arise in modern numerical relativity codes (Section 3). I then discuss numerical algorithms and codes for finding event horizons (Part II) and apparent horizons (Part III). Finally, in the appendices I briefly outline some of the excellent numerical algorithms/codes available for two standard problems in numerical analysis, the solution of a single nonlinear algebraic equation (Appendix A) and the time integration of a system of ordinary differential equations (Appendix B).

1 Notation and Terminology
2 2-Surface Parameterizations
 2.1 Level-set-function parameterizations
 2.2 Strahlkörper parameterizations
 2.3 Finite-element parameterizations
3 Software-Engineering Issues
 3.1 Software libraries and toolkits
 3.2 Code reuse and sharing
 3.3 Using multiple event/apparent horizon finders

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