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 similar to10(-5)m accuracy on a GHz-class processor. I assume that an AH to be searched for is a Strahlkorper ('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.