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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
A fast apparent-horizon finder for 3-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.
0264-9381
743-766
Thornburg, Jonathan
7164281d-b614-40e4-a27f-c89e874a7b9b
Thornburg, Jonathan
7164281d-b614-40e4-a27f-c89e874a7b9b

Thornburg, Jonathan (2004) A fast apparent-horizon finder for 3-dimensional Cartesian grids in numerical relativity. Classical and Quantum Gravity, 21 (2), 743-766. (doi:10.1088/0264-9381/21/2/026).

Record type: Article

Abstract

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.

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Published date: 2004

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Local EPrints ID: 45844
URI: http://eprints.soton.ac.uk/id/eprint/45844
ISSN: 0264-9381
PURE UUID: 94538afa-1586-420e-9b35-2aaac9bb8939

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Date deposited: 16 Apr 2007
Last modified: 26 Apr 2022 20:41

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Author: Jonathan Thornburg

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