Finding Apparent Horizons in Numerical Relativity

TL;DR

This paper introduces an efficient symbolic differentiation method for computing the Jacobian in Newton's method to find apparent horizons in numerical relativity, demonstrating rapid convergence and high accuracy.

Contribution

It presents a novel symbolic differentiation approach for Jacobian computation, improving efficiency and ease of implementation in horizon-finding algorithms.

Findings

Symbolic Jacobian computation is more efficient than numerical perturbation.

Newton's method converges rapidly with good initial guesses.

High-frequency errors in initial guesses reduce convergence radius.

Abstract

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced $H(h)$ function by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing $H(h)$. Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum $H(h)$ equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the $H(h)$ Jacobian to be {\em much} more efficient than the usual numerical perturbation method, and also much easier to implement than is commonly thought. When solving the discrete $H(h) = 0$ equations, we find that Newton's method generally converges very rapidly. However, if the initial guess for the horizon position contains significant high-spatial-frequency error components, Newton's method has a small (poor) radius of convergence. This is {\em not} an artifact of insufficient resolution in the finite difference grid; rather, it appears to be caused by a strong nonlinearity in the continuum $H(h)$ function for high-spatial-frequency error components in $h$. Robust variants of Newton's method can boost the radius of convergence by O(1) factors, but the underlying nonlinearity remains, and appears to worsen rapidly with increasing initial-guess-error spatial frequency. Using 4th~order finite differencing, we find typical accuracies for computed horizon positions in the $10^{-5}$ range for $\Delta\theta = \frac{\pi/2}{50}$.