Revisiting the apparent horizon finding problem with multigrid methods

Abstract

Apparent horizon plays an important role in numerical relativity as it provides a tool to characterize the existence and properties of black holes on three-dimensional spatial slices in 3+1 numerical spacetimes. Apparent horizon finders based on different techniques have been developed. In this paper, we revisit the apparent horizon finding problem in numerical relativity using multigrid-based algorithms. We formulate the nonlinear elliptic apparent horizon equation as a linear Poisson-type equation with a nonlinear source, and solve it using a multigrid algorithm with Gauss–Seidel line relaxation. A fourth order compact finite difference scheme in spherical coordinates is derived and employed to reduce the complexity of the line relaxation operator to a tri-diagonal matrix inversion. The multigrid-based apparent horizon finder developed in this work is capable of locating apparent horizons in generic spatial hypersurfaces without any symmetries. The finder is tested with both analytic data, such as Brill–Lindquist multiple black hole data, and numerical data, including off-centered Kerr–Schild data and dynamical inspiraling binary black hole data. The obtained results are compared with those generated by the current fastest finder AHFinderDirect (Thornburg 2003 Class. Quantum Grav.21 743), which is the default finder in the open source code Einstein Toolkit. Our finder performs comparatively in terms of accuracy, and starts to outperform AHFinderDirect at high angular resolutions ( ) in terms of speed. Our finder is also more flexible to initial guess, as opposed to the Newton’s method used in AHFinderDirect. This suggests that the multigrid algorithm provides an alternative option for studying apparent horizons, especially when high resolutions are needed.