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We have presented a program to develop a suite of standardized testbeds for
numerical relativity, and a first round of tests. All tests are specified in
great detail which extends to numerical methods, grid setup and choice of
output quantities to facilitate comparisons. The tests are based on vacuum
solutions and periodic boundaries. Even in this simple setup, the design of
tests is a highly nontrivial task and several subtle issues require careful
consideration: The effects of gravitational collapse introduce considerable
subtleties in tests of general relativity with periodic boundary conditions and
have been discussed in some detail. Comparing runs for spacetimes which possess
symmetries in different setups where the symmetry is manifest, disguised by a
coordinate transformation, or disguised by adding random noise can help to
understand the problem of what one can learn from simple one-dimensional tests,
and help test different aspects of a code. In particular, this has been found
useful in separating problems connected to ill-posedness from other sources of
instability or inaccuracy.
Several of the tests presented here have been used previously in one form or
another, but we have tried to improve their specifications in order to
increase their practical value. We have modified the robust stability test
based on random noise as presented in [Szilagyi et al.(2000)Szilagyi, Gómez,
Bishop, and Winicour] to reduce
computational resources when comparing different resolutions and added such a
comparison as an integral part of the test. Our setup of the collapsing
polarized Gowdy wave test combines a particularly simple choice of initial
data with a simple form of the exact solution.
The art of interpreting testbed results requires mastery of the art of
interpreting spacetimes. The latter has to be applied both to the continuum
limits and to the discretized approximations in order to understand results. A
simple example is provided by the gauge wave test, where individual runs may
exhibit collapse or expansion as a result of a physical instability of the
exact solution. Clearly, a valid code still has to show convergence to this
unstable exact solution. We strongly emphasize the importance of comparing
results for different resolutions. In particular, convergence tests not only
exhibit plain coding errors or numerical instabilities, but it is important to
obtain convergence information for all simulations individually, for the whole
length of a run. This is illustrated by our comparison of an ADM and a BSSN
code for the collapsing Gowdy test. Also, we emphasize that it is not
sufficient to monitor constraints to analyze instabilities, but further
quantities need to be analyzed to render possible scientifically valuable
conclusions.
We have carried out sufficient experimentation with these tests to ensure that
they can be implemented with reasonable computational resources and that they
can effectively discriminate between the performance of different codes. A
separate paper presenting and interpreting test results for codes of all groups
that wish to participate will be prepared at a later date. At present, we
invite all numerical relativity groups to submit results and join as co-authors
in this next paper.
Information on submitting results can be found at the present web site
www.ApplesWithApples.org. Instructions can also be found here for accessing
the results submitted by the various participating groups. We also encourage
groups to submit results from tests that go beyond the ones proposed here and
that reveal further insight into code performance. This would be particularly
helpful in the design of future tests. Also, information concerning
forthcoming workshops, and contact information for the participating groups,
are posted on the website.
The tests presented here are not intended to be an exhaustive or even
minimal list of tests that should be applied to a particular formulation or
code. However, they are sufficiently simple and general to allow all groups
to compare results with reasonable computational effort. They provide a way
of rapidly checking the utility of a code or formulation in situations
where detailed theoretical analysis is not possible. The tests also allow
isolation of problems of different origin, such as the mathematical
formulation, the choice of gauge or the inaccuracy of the numerical method.
They do this in a simple situation where cross-comparison with other codes
can suggest remedies.
We are proposing here the first step toward establishing a community wide
resource which will allow all groups to profit from each other's successes
and failures. Broad participation is essential to the success of this goal.
Future workshops, along the lines of the first Mexico workshop, are being
planned. The key challenge for the next round of tests will be to
include the significantly more complex problem of boundaries.
- Szilagyi et al.(2000)Szilagyi, Gómez,
Bishop, and Winicour
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B. Szilagyi,
R. Gómez,
N. T. Bishop,
and J. Winicour,
Phys. Rev. D 62
(2000), gr-qc/9912030.
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