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Test case specification

Periodic boundaries

Boundaries

Comments

We will specify the physical properties of each testbed by providing the complete 4-metric of the spacetime, or if this is not possible, the initial Cauchy data and choice of gauge for the evolution. In all cases, we will give the Cauchy data, i.e. the 3-metric and extrinsic curvature, in a Cartesian coordinate system appropriate for 3-dimensional evolution. The physical domain is a cube which, in this first round of tests with periodic boundary conditions, represents a 3-torus.

In order for uniform comparison the series of four tests should be run using a second order iterative-Crank-Nicholson algorithm with two iterations (in the notation of [Teukolsky(2000)]) with second order accurate finite differencing in space. There may be codes that cannot implement this type of numerical method. Similarly, a particular code may run better with an alternative numerical method such as a Runge-Kutta time integrator or a pseudo-spectral method. In such cases the relative performance of the code for these tests still offers a useful comparison, provided all parameters (such as the amount of artificial dissipation) are held constant over the four tests. However, for a quantitative cross comparison of codes it is best to provide results from a standard numerical method. Second order iterative-Crank-Nicholson is chosen for simplicity.

The simulation domain for each test will generally be a cube of side $d$, equal to one wavelength with periodic boundary conditions. The grids are set up to extend an equal distance in the positive and negative directions along each axis. As depicted in Fig. 1, the ``boundaries'', which are identified in the 3-torus picture, are located a half step from the first and last grid points along each axis. The resolution in a direction $i$ is given by $\Delta x^i = d/n^i$. The number of grid points $n^i$ should be sufficient to resolve features of the initial data in the given direction. Even though we are running three-dimensional codes, for tests with only one-dimensional features it is considerably more efficient to restrict the grid such that $n^i$ is small in the trivial directions. As an example, for a wave propagating in the $x$-direction we use the minimum number of grid points in the trivial $y$ and $z$ directions that allow for non-trivial numerical second derivatives. For standard second order finite differencing this implies that we use 3 points in those directions.

Figure 1: Grid points (in this case n=8) along a given axis are chosen to straddle both $x=0$ and the identified boundaries. An arbitrary number of ghost-zone points beyond the boundaries can be used in implementing periodic boundary conditions.
\includegraphics[width=30pc]{grid.eps}

The size of the timestep $dt$ is given in terms of the grid size $dx$ and chosen to lie within the CFL limit for an explicit evolution algorithm. We foresee the possibility of codes for which this would be inappropriate and for which some equivalent choice of time step would have to be made. A final time $T$, and intermediate times for data output, are specified for each test. The time $T$ is chosen to incorporate all useful features of the test without prohibitive computational expense. The output times should be appropriately modified for codes that crash before time $T$.

The output quantities are chosen with either some physical or numerical motivation in mind. Both quantitative and qualitative comparisons are used. Some examples, based upon representative versions of evolution codes, are provided below for purposes of illustration. We specify a minimum list of output quantities. Other quantities that we do not include here but that may be of interest for a specific code include the Fourier transform of differences between the numerical and exact solutions as in [Calabrese et al.(2002)Calabrese, Pullin, Sarbach, and Tiglio], curvature invariants to detect deviations from flat space, and proper time integrated along observer world lines. Each group should of course also output any additional variables which are essential to the performance of their particular formulation.

Bibliography

Teukolsky(2000)
S. Teukolsky, Phys. Rev. D 61, 087501 (2000).
Calabrese et al.(2002)Calabrese, Pullin, Sarbach, and Tiglio
G. Calabrese, J. Pullin, O. Sarbach, and M. Tiglio, Phys. Rev. D 66, 041501 (2002), gr-qc/0207018.
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$Date: 2006/11/30 16:07:26 $