A fast apparent horizon finder for three-dimensional Cartesian grids in numerical relativity*

In 3 + 1 numerical simulations of dynamic black-hole spacetimes, it is useful to be able to find the apparent horizon(s) (AH) in each slice of a time evolution. A number of AH finders are available, but they often take many minutes to run, so they are too slow to be practically usable at each time step. Here I present a new AH finder, AHFINDERDIRECT, which is very fast and accurate: at typical resolutions it takes only a few seconds to find an AH to ∼10−5m accuracy on a GHz-class processor. I assume that an AH to be searched for is a Strahlkörper (‘star-shaped region’) with respect to some local origin, and so parametrize the AH shape by r = h (angle) for some single-valued function h : S2 → +. The AH equation then becomes a nonlinear elliptic PDE in h on S2, whose coefficients are algebraic functions of gij ,Kij , and the Cartesian-coordinate spatial derivatives of gij . I discretize S2 using six angular patches (one each in the neighbourhood of the ±x,±y, and ±z axes) to avoid coordinate singularities, and finite difference the AH equation in the angular coordinates using fourth-order finite differencing. I solve the resulting system of nonlinear algebraic equations (for h at the angular grid points) by Newton’s method, using a ‘symbolic differentiation’ technique to compute the Jacobian matrix. AHFINDERDIRECT is implemented as a thorn in the CACTUS computational toolkit, and is freely available by anonymous CVS checkout. PACS numbers: 04.25.Dm, 02.70.Bf, 02.60.Cb M This article features online multimedia enhancements (Some figures in this article are in colour only in the electronic version) * Appendix B on ‘multiprocessor and parallelization issues’ and appendix C on ‘searching for the critical parameter of a 1-parameter initial data sequence’ also appear in the preprint-archive version of this paper (gr-qc/0306056). 0264-9381/04/020743+24$30.00 © 2004 IOP Publishing Ltd Printed in the UK 743