Christiane Lechner

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations ✩

We present a suite of Mathematica-based computer-algebra packages, termed “Kranc”, which comprise a toolbox to convert certain (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code for solving initial boundary value problems. Kranc can be used as a “rapid prototyping” system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations. Program summary

Towards standard testbeds for numerical relativity

In recent years, many different numerical evolution schemes for Einsteins 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 Einsteins 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 towards 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.

Implementation of standard testbeds for numerical relativity

We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.

Type II Critical Collapse of a Self-Gravitating Nonlinear Sigma Model

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) σ models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant η ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period Δ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent γ which is almost independent of η, asymptoting to 0.1185±0.0005 at large η. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing–null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters show second order convergence with the grid resolution, and after compensating for this variation in our individual evolutions are uniformly second order convergent even very close to criticality.

σmodel on de Sitter space

We discuss spherically symmetric, static solutions to the SU(2) sigma model on a de Sitter background. Despite of its simplicity this model reflects many of the features exhibited by systems of non-linear matter coupled to gravity e.g. there exists a countable set of regular solutions with finite energy; all of the solutions show linear instability with the number of unstable modes increasing with energy.