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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 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
B. Szilagyi, R. Gómez, N. T. Bishop, and J. Winicour, Phys. Rev. D 62 (2000), gr-qc/9912030.
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