Test case specification
Periodic boundaries
Boundaries
Comments
We will specify the physical properties of each testbed by providing
the complete 4metric 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 3metric and extrinsic
curvature, in a Cartesian coordinate system appropriate for
3dimensional evolution. The physical domain is a cube
which, in this first round of tests with periodic boundary conditions,
represents a 3torus.
In order for uniform comparison the series of four tests should be run
using a second order iterativeCrankNicholson 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
RungeKutta time integrator or a pseudospectral 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
iterativeCrankNicholson is chosen for simplicity.
The simulation domain for each test will generally be a cube of side ,
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 3torus picture, are located
a half step from the first and last grid points along each axis. The
resolution in a direction is given by
. The
number of grid points should be sufficient to resolve features of
the initial data in the given direction. Even though we are running
threedimensional codes, for tests with only onedimensional features it
is considerably more efficient to restrict the grid such that is
small in the trivial directions. As an example, for a wave propagating
in the direction we use the minimum number of grid points in the
trivial and directions that allow for nontrivial 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 and
the identified boundaries. An arbitrary number of ghostzone
points beyond the boundaries can be used in implementing
periodic boundary conditions.

The size of the timestep is given in terms of the grid
size 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 , and intermediate
times for data output, are specified for each test. The time 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 .
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), grqc/0207018.
