In 3+1 numerical simulations of dynamic black hole spacetimes, it's useful tobe 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 presenta new AH finder,_AHFinderDirect_, which is very fast and accurate: at typicalresolutions 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-shapedregion) 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 AHequation then becomes a nonlinear elliptic PDE in $h$ on $S^2$, whosecoefficients are algebraic functions of $g_{ij}$, $K_{ij}$, and theCartesian-coordinate spatial derivatives of $g_{ij}$. I discretize $S^2$ using6 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 AHequation in the angular coordinates using 4th order finite differencing. Isolve the resulting system of nonlinear algebraic equations (for $h$ at theangular grid points) by Newton's method, using a "symbolic differentiation"technique to compute the Jacobian matrix._AHFinderDirect_ is implemented as athorn in the_Cactus_ computational toolkit, and is freely available byanonymous CVS checkout.
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机译：在动态黑洞时空的3 + 1数值模拟中，找到时间演化的每个切片中的视在视界（AH）很有用。许多AH发现者可用，但它们通常要花几分钟才能完成运行，因此它们太慢而无法在每个时间步上实际使用。在这里，我介绍了一个新的AH查找器_AHFinderDirect_，它非常快速且准确：在典型的分辨率下，只需几秒钟即可在GHz级处理器上找到达到$ \ sim 10 ^ {-5} m $精度的AH。我假设要搜索的AH是相对于某些本地原点的Strahlk \“ perper（星状区域），因此对于某些单值函数$ h，通过$ r = h（angle）$来对AH形状进行参数化：S ^ 2 \ to \ Re ^ + $。然后，AHequation变成$ h $在$ S ^ 2 $上的非线性椭圆PDE，其系数是$ g_ {ij} $，$ K_ {ij} $，以及$ g_ {ij} $的笛卡尔坐标空间导数。我使用6个角度斑块（在$ \ pm x $，$ \ pm y $和$ \ pm z $附近分别一个）对$ S ^ 2 $进行离散化。轴坐标），以避免坐标奇异，并使用四阶有限差分法对角坐标中的AH方程进行有限差分。通过牛顿法，使用“符号微分”，求解非线性代数方程组（在角网格点处为$ h $）。 _AHFinderDirect_在_Cactus_计算工具箱中作为尖刺实现，并且可以通过匿名CVS结帐免费获得。
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