A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity

Jonathan Thornburg

doi:10.1088/0264-9381/21/2/026

paper

A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity

arXiv:gr-qc/0306056 · doi:10.1088/0264-9381/21/2/026

Abstract

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 $\sim 1 0^{- 5} m$ 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 parameterize the AH shape by $r = h (an g l e)$ for some single-valued function $h : S^{2} \to ℜ^{+}$ . 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 6 angular patches (one each in the neighborhood of the $\pm x$ , $\pm y$ , and $\pm z$ 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.

26 pages + 3 page supplement, LaTeX2e, RevTeX4, includes color postscript figures; changes from v2: revised based on referee comments, several nasty typos fixed, one appendix deleted and two others split off into a supplemental file

A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity

Abstract

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