Topics in numerical relativity: Solving the initial value problem using adaptive mesh refinement, examining evolution stability using spectral methods, and finding apparent horizons using a mean curvature-level set method

摘要

Various topics in numerical relativity will be discussed, including solving the initial value problem using adaptive mesh refinement, examining evolution stability using spectral methods, and finding apparent horizons using a mean curvature-level set method. We use the ADM 3+1 formalism, which separates the Einstein equations into four constraint equations and twelve evolution equations. The four constraint equations are then transformed via a York-Lichnerowicz transverse traceless decomposition. We give a brief discussion on variations of this decomposition. We discuss the implementation of a parallel multigrid solver with adaptive mesh refinement for solving the initial value constraint equations. Using our multigrid solver, we solve for a new class of initial data: distorted black holes using the puncture method. The ADM evolution equations have known instabilities. We explore the instabilities inherent in the evolution equations with an evolution code implemented using spectral methods with a Runge-Kutta integrator via the method of lines. An overview of spectral methods is given. We compare the results of evolving the full ADM evolution equations with stability predictions from integrating the constraint equations. Evolution instabilities can be contained by adding multiples of the constraints to the evolution equations. We show how to implicitly dictate the behavior of constraint violation during an evolution by manipulating additional constraint terms. After solving for initial data or during an evolution, it is often necessary to locate the position of the apparent horizon. We rewrite the apparent horizon equation as a surface evolving along its normal vector according to the speed of the apparent horizon equation. Instead of evolving a two dimensional surface and assuming a star shaped topology, we evolve a three dimensional surface via the level set method. This allows us to find multiple apparent horizons of any topology in generic spacetimes, either analytic or numerically generated.