We present results of 3D numerical simulations using a finite difference code featuring fixed mesh refinement (FMR), in which a subset of the computational domain is refined in space and time. We apply this code to a series of test cases including a robust stability test, a nonlinear gauge wave and an excised Schwarzschild black hole in an evolving gauge. We find that the mesh refinement results are comparable in accuracy, stability and convergence to unigrid simulations with the same effective resolution. At the same time, the use of FMR reduces the computational resources needed to obtain a given accuracy. Particular care must be taken at the interfaces between coarse and fine grids to avoid a loss of convergence at higher resolutions, and we introduce the use of "buffer zones" as one resolution of this issue. We also introduce a new method for initial data generation, which enables higher-order interpolation in time even from the initial time slice. This FMR system, "Carpet", is a driver module in the freely available Cactus computational infrastructure, and is able to endow generic existing Cactus simulation modules ("thorns") with FMR with little or no extra effort.

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In recent years, many different numerical evolution schemes for Einstein's equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches have been tested on different spacetimes, and conclusions have been drawn based on these tests. However, differences in results originate from many sources, including not only formulations of the equations, but also gauges, boundary conditions, numerical methods, and so on. We propose to build up a suite of standardized testbeds for comparing approaches to the numerical evolution of Einstein's equations that are designed to both probe their strengths and weaknesses and to separate out different effects, and their causes, seen in the results. We discuss general design principles of suitable testbeds, and we present an initial round of simple tests with periodic boundary conditions. This is a pivotal first step toward building a suite of testbeds to serve the numerical relativists and researchers from related fields who wish to assess the capabilities of numerical relativity codes. We present some examples of how these tests can be quite effective in revealing various limitations of different approaches, and illustrating their differences. The tests are presently limited to vacuum spacetimes, can be run on modest computational resources, and can be used with many different approaches used in the relativity community.

We present a study of black hole threshold phenomena for a self-gravitating, massive complex scalar field in spherical symmetry. We construct Type I critical solutions dynamically by tuning a one-parameter family of initial data composed of a boson star and a massless real scalar field. The real field is used to perturb the boson star via a gravitational interaction which results in a {\em significant} transfer of energy. The resulting critical solutions, which show great similarity with unstable boson stars, persist for a finite time before dispersing or forming a black hole. We extend the stability analysis of Gleiser and Watkins [Nucl. Phys. B319, 733 (1989)], providing a method for calculating the radial dependence of boson star modes of nonzero frequency. We find good agreement between our critical solutions and boson star modes. For critical solutions less than 90% of the maximum boson star mass $M_{\rm max} \simeq 0.633 M_{Pl}^2/m$, a small halo of matter appears in the tail of the solution. This halo appears to be an artifact of the collision between the original boson star and the real field, and does not belong to the true critical solution. It seems that unstable boson stars are unstable to dispersal in addition to black hole formation. Given the similarity in macroscopic stability between boson and neutron stars, we suggest that neutron stars at or beyond the point of instability may also be unstable to explosion.Comment: 26 Pages, 16 Figures, RevTeX. Submitted to Phys. Rev.