# AHFinderDirect

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 -5 m 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:S 2 →ℜ + . The AH equation then becomes a nonlinear elliptic PDE in h on S 2 , whose coefficients are algebraic functions of g ij , K ij , and the Cartesian-coordinate spatial derivatives of g ij . I discretize S 2 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.

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## References in zbMATH (referenced in 21 articles )

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