In the 1990s, some researchers were concentrating on obtaining physics insight from effectively 1+1 dimensional problems: what cosmological spacetimes with two commuting Killing vectors can tell us about the nature of generic singularities (Berger and collaborators), and what we can learn about cosmic censorship from spherical collapse (Choptuik and students). More ambitious, axisymmetric or 3D, work confronted overlapping problems hard to disentangle in the low resolution available in 3D. In particular, instabilities already present in the continuum problem were not clearly distinguished from those added at the discretization stage. The Banff meeting showed that now at least we have a clearer view of the problems facing us.
3+1 approaches need to start from a well-posed initial-boundary value problem in the continuum, with boundary conditions that are compatible with the constraints. Well-posedness can be proved by energy methods, based on a symmetric hyperbolic form of the field equations. Olivier Sarbach drops the energy estimate based on the symmetrizer in favor of a ``physical'' energy plus a constraint energy. The remaining ``gauge'' energy is estimated separately using elliptic gauge conditions. This intuitively appealing programme has been completed for electromagnetism, although the gauge seems a bit restrictive. Work with Nagy is under way on general relativity. By contrast Oscar Reula emphasized that strong hyperbolicity is often enough. He could prove that whenever a first-order system subject to constraints is strongly hyperbolic (eg the BSSN formulation) then so is the associated constraint evolution system. Heinz Kreiss surprised some of his disciples in the numerical relativity community by also stressing that energy methods are too limited. In a series of examples, he proposed a general approach based on reducing initial-boundary value problems to half-space problems with frozen coefficients and analyzing the dependence of each Fourier mode on its initial and boundary data.
On the numerical methods front, Manuel Tiglio reported on collaborative work to discretize systems of first-order strongly hyperbolic equations on multiple touching patches (for example 6 cubes to form a hollow sphere), using summation by parts and penalty methods. Their animations of toy problems looked very impressive, and the whole technology will be available as a general tool through the Cactus infrastructure. Michael Holst and Rick Falk gave review talks on finite elements for both elliptic and evolution equations. This is promising for nontrivial domains, but has not yet been applied to numerical relativity.
Other talks showed what 3D simulations can do. David Garfinkle reported on simulations of cosmological singularities without any symmetries on . The key elements of his approach are the use of inverse mean curvature flow slicing () and a tetrad and connection formulation used successfully by Uggla and coworkers in analytical studies. His results are compatible with the BKL conjecture, although soon the resolution becomes too low to follow the development of ever more decoupled Bianchi IX regions. Thomas Baumgarte summarized the state of the art in binary neutron star simulations by himself and others, notably Masaru Shibata. There seems to be no real showstopper for such simulations. Rather what is needed now is more resolution, and the modelling of physical phenomena such as neutrinos, viscosity, and magnetic fields. Interesting results include the formation in binary mergers of a hot neutron star held up only by differential rotation, and expected to collapse later.
The most noted talk of the meeting was that of Frans Pretorius
giving preliminary results on binary black hole mergers using harmonic
coordinates. His simulations no longer seem to be limited by
instabilities, but rather by computer power and time, and by
unphysical initial data (there is evidence that his initial data are
very far from circular inspiral data). The key ingredients seem to be
the following: a working 3D AMR code on still massive computers,
compactification of the Cartesian spatial coordinates (that is, at
) together with damping of outgoing waves, modified harmonic
coordinates, and a damping of the harmonic gauge constraint through
lower order friction terms (Gundlach). Generalized
harmonic gauge (Friedrich) is
, where the gauge
source functions are treated as given functions. Pretorius
makes obey a wave equation
prevents the lapse from collapsing without affecting the
well-posedness. This works less well for critical collapse.