Modeling the Black Hole Excision Problem

TL;DR

This paper develops and analyzes a numerical modeling approach for simulating black hole excision, ensuring well-posedness and stability, and demonstrating accurate long-term simulations applicable to Einstein's equations.

Contribution

It introduces a second-order differential treatment and boundary algorithms for black hole excision, with proofs of well-posedness and suppression of exponential modes.

Findings

Accurate long-term simulations of quasi-linear black hole excision.

Proofs of well-posed evolution and boundary algorithms.

Suppression of long wavelength exponential modes.

Abstract

We analyze the excision strategy for simulating black holes. The problem is modeled by the propagation of quasi-linear waves in a 1-dimensional spatial region with timelike outer boundary, spacelike inner boundary and a horizon in between. Proofs of well-posed evolution and boundary algorithms for a second differential order treatment of the system are given for the separate pieces underlying the finite difference problem. These are implemented in a numerical code which gives accurate long term simulations of the quasi-linear excision problem. Excitation of long wavelength exponential modes, which are latent in the problem, are suppressed using conservation laws for the discretized system. The techniques are designed to apply directly to recent codes for the Einstein equations based upon the harmonic formulation.