  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.
