# Testing outer boundary treatments for the Einstein equations

## TL;DR

This paper compares various outer boundary treatments in numerical relativity, assessing their effectiveness in minimizing reflections and constraint violations using a Schwarzschild black hole test case.

## Contribution

It introduces an improved boundary condition for gauge degrees of freedom and evaluates multiple boundary treatments within a unified framework.

## Key findings

- Constraint-preserving boundary conditions perform well in reducing reflections.
- Freezing the Newman-Penrose scalar Psi_0 minimizes boundary reflections.
- Combining constraint preservation with gauge control yields the best results.

## Abstract

Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a `reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests use a first-order generalized harmonic formulation of the Einstein equations. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar Psi_0, and control gauge reflections.