``Apples with Apples no Oranges''

Workshop on Formulations of Einstein's Equations

for Numerical Relativity

Miguel Alcubierre, UNAM, Mexico malcubi@nuclecu.unam.mx

Late Spring saw a very successful workshop concentrating on formulations of the Einstein equations for numerical relativity. The workshop was held at Institute of Nuclear Sciences of the National University of Mexico (UNAM) in Mexico city on May 13-24, and was attended by some 25 people from Mexico, the U.S. and Europe (with a particularly large group from Germany). The purpose of the workshop was to gather a group of experts in the recent developments of the different formulations of the Einstein equations and their applications to numerical relativity. There was a focus in 3+1 formulations, though the conformal approach was also represented. One of the highlights of the meeting was our trip to Teotihuacan, where many a pyramid was climbed and where the god of rain was kind on us by throwing torrential rains during our bus trip to refresh the place, and stopping just as we arrived. But we also talked about physics, of course.

The workshop was motivated by the growing realization in the numerical relativity community that different forms of writing the evolution equations can have tremendous impact in the long term stability and accuracy of a numerical simulation. Some formulations have been show to be either simply ill-posed in a mathematical sense, or else such that well-posedness can not be proved. Such is the case, for example, of the standard ADM formulation. Formulations that can be shown to be well-posed, for example strongly hyperbolic ones, should clearly have an advantage.

During the workshop several key points became clear: hyperbolicity, though desirable, is not enough, as several groups reported that some strongly hyperbolic formulations are far superior than others. Also, hyperbolicity might not even be necessary for long-term stability, as some formulations that are not directly hyperbolic (as the BSSN formulation) have been shown to be remarkably robust. Moreover, recent developments, in particular the introduction of multiple-parameter families of strongly hyperbolic formulations by Kidder et all (the KST family), make it clear that simply comparing many formulations against each other in direct numerical experiments is a difficult task, and theoretical insights are badly needed to help us chose the more promising formulations from an everyday growing zoo.

The meeting lasted two weeks, and was organized with talks in the morning and working sessions in the afternoon during the first week, and informal discussions plus more working sessions the second week. It is difficult to discuss all contributions to the workshop here, but I will make an attempt to mention a few highlights:

On Monday, Lee Lindblom talked about the recent KST family of strongly hyperbolic formulations, and mentioned what I believe to be a very important development in the field, the realization that since symmetric hyperbolic formulations allow a norm to be constructed, one can in fact predict ahead of time the rate of growth of solutions on a given background. This has led to the discovery that this growth is essentially a linear phenomenon, so analysis of linearized equations around non-trivial backgrounds should be sufficient. Carles Bona then presented a framework under which many different formulations, from the ADM and BSSN formulations to the Bona-Masso and KST formulations can be studied under a unified approach.

On Tuesday, Hisaaki Shinkai showed the results of many tests he has carried out with many formulations, and presented the hypothesis that a Fourier mode analysis of linearized equations around specific backgrounds should be a good indicator of which formulation will behave better than others. Thomas Baumgarte presented a beautiful analysis of electrodynamics where he showed how one can construct an analogous of the BSSN formulation in relativity and see very clearly what the benefits of such a form of the evolution equations are. Erik Schnetter talked about ways in which different gauge conditions can and should be enforced during a numerical simulation. Finally, Deirdre Shoemaker talked about recent advances that Pablo Laguna's group at PSU has had in developing formulations based on BSSN, but where care is taken to eliminate some quadratic source terms. Their approach is to look systematically not only at the principal part, but also a lower order terms in the search for better stability.

Wednesday saw the talks representing the conformal approach, given by Christiane Lechner and Sascha Husa. Their approach, based on Friedrich's conformal equations is capable of evolving all the way to null infinity (and beyond), seems to have all the benefits of the Cauchy-characteristic matching and fewer of its problems. Still, at this time it also seems to have problems with long term stability. Wednesday also saw our trip to Teotihuacan.

On Thursday Jeff Winicour and Bela Szilagyi reminded us all of the importance of choosing adequate boundary conditions that are designed to satisfy the constraint equations. Their efforts on developing consistent boundary conditions are a model of what all other groups should be worrying about when writing Cauchy codes. The interaction of boundaries and interior should clearly be taken into account when thinking about the long-term stability of the different codes. Ryoji Takahashi also talked about gauge stability of different 3+1 formulations.

Friday saw a talk by Miguel Alcubierre presenting the stability analysis of BSSN in linearized gravity. Denis Pollney talked about the recent experience of the Potsdam group with BSSN in black hole simulations, and Ian Hawke gave a general talk on how to implement in a simple and systematic way a general method of lines algorithm within the Cactus framework.

Working sessions concentrated in getting the different codes working together under a similar framework. Whenever possible, different codes where ported into the Cactus framework to facilitate sharing initial data and analysis tools. The meeting was very successful, and people have continued to work together several months afterward trying to compare results of the different approaches in specific situations.

As a final personal comment I would like to add that the workshop had also an impact in the Institute of Nuclear Sciences in Mexico, where I work. Initially, a computer terminal room was booked for the working sessions, but it quickly became apparent that laptops and wireless networks could be used to transform our main lecture theater into a much better working environment. The sight of 20 plus people having claimed the main auditorium for two weeks solid, typing away at their laptop keyboards while simultaneously sharing data through the Internet was enough to impress everyone, and in my view showed what should probably become a standard in future numerical relativity workshops.

Jorge Pullin 2002-09-23