{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-fast-apparent-horizon-finder-for-3","title":"A Fast Apparent-Horizon Finder for 3-Dimensional Cartesian Grids in Numerical Relativity","arxiv_id":"gr-qc/0306056","date":"2003-06-12","proceeding":null,"authors":["Jonathan Thornburg"],"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 $\\sim 10^{-5} m$ 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 $h: S^2 \\to \\Re^+$. 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.","url_abs":"https://arxiv.org/abs/gr-qc/0306056v3","url_pdf":"https://arxiv.org/pdf/gr-qc/0306056v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-fast-apparent-horizon-finder-for-3","repo_url":"https://github.com/cwru-pat/GR_AMR","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}