Papers › A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity
A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity
Jonathan Thornburg
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In 3+1 numerical simulations of dynamic black hole spacetimes, it's 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're 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 Strahlk\"orper (star-shaped region) with respect to some local origin, and so parameterize 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 6 angular patches (one each in the neighborhood of the ±x, ±y, and ±z axes) to avoid coordinate singularities, and finite difference the AH equation in the angular coordinates using 4th 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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