Numerical Relativity Workshop at IMA

Manuel Tiglio, Louisiana State University tiglio@lsu.edu

The Numerical Relativity (NR) Workshop organized by the Institute for Mathematics and its Applications (IMA) at the University of Minnesota and jointly sponsored with the Center for Gravitational Wave Physics (CGWP) at Penn State, took place in IMA's facilities from June 24th to June 29th (the last week of the 2002 Soccer World Cup, a detail not ignored by many of the attendees).

As announced, the workshop effectively ``brought together numerical relativists and mathematicians working in fields such as numerical analysis, scientific computation, partial differential equations and geometry, for an intense but informal period aimed at maximal communication and interaction between diverse researchers''. Considerable effort was put in bringing these communities closer. Relativists tried to describe the status and problems of NR and differences between General Relativity (GR) and Maxwell theory or fluid dynamics. On the other hand, numerical analysts made their best to bring their expertises closer to our field. At the end of the workshop the latter were much more familiar with the problems of NR, while relativists got to learn and revisit a plethora of mathematical techniques that may help solve some of the problems.

On the first day mathematicians were introduced to GR by talks given by Doug Arnold (director of IMA), Alan Rendall (AEI), and Robert Bartnik (University of Canberra). Arnold gave a self consistent tutorial to Einstein's equations from a mathematician's point of view. Followed Rendall, discussing the $3+1$ decomposition and local existence and uniqueness to the Cauchy problem. Bartnik, in turn, explained slicing conditions, the constraint equations and gravitational radiation.

On Tuesday morning, Ralf Hiptmair (Universität Bonn) reviewed discretizations of Maxwell's equations, putting emphasis on coordinate- free methods. He included the finite volume approach, generalized finite differences, and finite elements. Later, Eitan Tadmor (University of Maryland) presented discretizations for nonlinear hyperbolic systems that preserve local and global invariants. Since these discretizations are motivated mainly by computational fluid dynamics, there were several discussions on how to make use of these techniques in Einstein's vacuum evolutions. On Thursday morning Tadmor continued the discussion with some remarks about hyperbolic formulations and giving some ideas to diminish the accumulation errors by trading some hyperbolic equations for elliptic ones.

Back on Tuesday, during the afternoon Oscar Reula (University of Córdoba) talked about the role of hyperbolicity in formulations of Einstein's equations. He discussed the weak-ill posedness of the standard ADM formulation and the ``hidden'' hyperbolicity of BSSN-like systems. He also mentioned the linear degeneracy of usual formulations of Einstein's equations. Markus Keel (IMA) on Thursday morning gave a talk further elaborating this point and others. Namely, he explained that usual formulations for Einstein's vacuum equations are not genuinely nonlinear and therefore one should not expect shocks. He also discussed the ``stability problem'' in NR (later in the workshop to be covered in more detail by Lindblom and Scheel), and a stability result by Kreiss, Ortiz and Reula for hyperbolic systems. Going back once again to Tuesday, after Reula's talk I gave one overviewing NR as an initial-boundary value problem, summarizing what is known about constraint preservation, well posedness, and numerical stability for finite difference schemes in the presence of boundaries.

On Wednesday morning, Jeff Winicour (University of Pittsburgh) gave a one hour and a half introduction to black holes (BHs), covering topics such as conformally compactified spacetimes, the notion of an event horizon and its intrinsic geometry, the no-hair hypothesis, and a description of gravitational collapse. Later in the morning, Matt Choptuik (University of British Columbia) talked about fundamental issues in NR. Among other things, he discussed BH excision, coordinate conditions, optimization, differences with other kind of numerical computations, adaptative mesh refinement, gravitational collapse, and boundary conditions. He pointed out something that was going to be mentioned by mathematicians several times during the workshop as well. Namely, that while the majority of NR implementations are obtained with free (unconstrained) evolution, there are schemes for solving elliptic equations (e.g. multigrid methods) where the number of operations scales linearly with number of gridpoints. As highlights, he discussed the issue of choosing good model problems, Brandt's golden rule that the amount of numerical computation should be proportional to the amount of real physical changes in the computed system, and that the situations for which we most need NR are those for which there is little a priori knowledge.

Pablo Laguna (PSU) gave the last talk on Wednesday afternoon, giving an overview of the state of the art in NR. He reviewed $3$D evolutions of single BHs, BH-BH, BH-neutron stars (NSs), and NS-NS binaries, and $2$D simulations of gravitational collapse. He explained what formulations of Einstein's equations are used in these simulations, and the available results.

Later in the afternoon there was a pizza dinner followed by computational demonstrations. Winicour presented numerical evolutions of a well posed initial-boundary formulation of vacuum GR using the harmonic gauge, and some comparison tests from the Workshop on Formulations of Einstein equations for NR ² that showed improved stability in this formulation. Jincha Xu (Penn State) discussed problem-independent adaptivity and multigrid methods. Michael Holst (University of California at San Diego) presented the (finite element) Manifold Code (MC) developed at UCSD, designed to solve nonlinear elliptic systems. Bartnik showed quasi-spherical numerical evolutions of BHs via a characteristic formulation. Dennis Pollney (AEI) presented $3$D simulations using new coordinate conditions that they applied to binary BH evolutions, with improved stability properties.

Thursday morning was devoted to the initial data problem, with two talks, given respectively by Greg Cook (Wake Forest University) and Holst. Cook reviewed the 3+1 decomposition, the conformal and physical transverse-traceless decompositions, and the conformal thin sandwich approach. Then, he described BHs and NSs initial data. He ended pointing out current trends, such as initial data with more astrophysical content and/or data for objects in quasi equilibrim. Holst's talk was a natural continuation of Cook's, this time addressing mathematical issues. For example, he overviewed existence results for the Hamiltonian constraint, as well as more recent techniques for the coupled Hamiltonian-Momentum equations. He gave a detailed discussion of well posedness, a priori and a posteriori error estimates, and then moved to numerical solutions, mainly using finite element methods. He also presented some specific results for BH initial data.

In the afternoon, Deirdre Shoemaker (PSU) talked about BH excision and apparent horizon tracking. She discussed different numerical methods for this tracking, as well as the different discretizations that are currently used for finite differencing at the inner boundary in the case of excision. As an application she presented the status of simulations of moving $3$D Schwarzschild BH. Followed Mark Scheel's (Caltech) talk about pseudospectral evolution of BHs. He explained pseudo spectral collocation methods and then discussed the constraint violating problem in NR (namely, fast growing, non-physical modes excited by numerical errors). He showed several evolutions of a $3$D Schwarzschild BH, emphasizing the dependence of the stability on the choice of the formulation of Einstein's equations. He also discussed a priori analytical energy estimates, a point that was later elaborated in more detail by Lee Lindblom (Caltech) on Friday. In his talk Lindblom explained their method to obtain sharp estimates, which can be used to a priori choose a formulation that optimizes the stability of a given background, and presented numerical results that confirmed their analytical predictions.

On Friday morning, Luis Lehner (University of British Columbia) gave the state of the art of outer boundary conditions in NR. He discussed standard and new ways of handling boundaries in Cauchy evolution, and the characteristic and conformal approaches. He also stressed that one can actually make use of Gustaffson-Kreiss-Sundstrum-Osher's theory in NR to construct stable boundary treatments. Richard Falk (Rutgers University) explained finite element methods for hyperbolic equations, how to parallel continuum estimates in the design of numerical methods, and different mesh constructions. Sascha Husa (AEI) summarized the conformal approach to NR, discussing previous work (e.g. constructing the whole spacetime describing weak gravitational waves) and some needs (for example, live gauges) and difficulties . He emphasized that many of these difficulties also appear in other, more standard, approaches to NR.

Sam Finn and Lee Lindblom closed Friday's presentations with two talks. During the week, computational scientists had asked how gravitational waves are related to solutions of Einstein's equations. Finn addressed these questions with a talk that explained the basics of laser interferometry, and generation and observation of gravitational waves. Lindblom talked about simulation of r-modes through pseudo-Newtonian models.

During the afternoon, there was a panel discussion on numerical methods. Topics such as adaptative mesh refinement and the possibility of general purpose tools were discussed. The possibility of explicitly solving the constraints during evolution was also brought up, especially given that several of the computational scientists present were experts in solving elliptic equations with advanced, efficient methods. The current lack of understanding in NR of boundary conditions for hyperbolic-elliptic systems was in turn presented by numerical relativists as one of the main obstacles in implementing such methods.

The workshop ended on Saturday noon, after a discussion lead by Richard Price. He presented the ``many aspects of the elephant'', and an original yin-yang summary of different opinions about to attempt obtaining gravitational wave templates in the short term (we are doing fine/we need a new idea, specific focus/all areas, careful analysis/computer experiments, more intuition/more math, big codes/model problems, etc) that was used to start a general discussion of what would be the best way to proceed.

People agreed on continuing the workshop in the future with other meetings. Doug Arnold suggested joining together smaller groups of people willing to collaborate or to discuss more specific aspects. Sam Finn, as director of the CGWP, offered the Center as another natural place for these future meetings. As one can imagine from the number of talks given and variety of topics discussed, the workshop was very intense. People went back home with new ideas, renewed hopes, and just in time to see Brazil win the final game of the World Cup. The conference website is http://www.ima.unm.edu/nr