(ed.). Gravitational waves (IOP, 2001)(422s).

392 Numerical relativityinitial distribution may not even see the final BH. Further, as pulses of radiationpropagate back out from the origin, they too may have to be resolved in regionswhere there was previously a coarse grid. Choptuik’s AMR system, built on earlywork of Berger and Oliger [138], was able to track dynamically features thatdevelop, enabling him to discover and accurately measure BH critical phenomenathat have now become so widely studied [139].Based on this success and others, and on the general considerations discussedabove, full 3D AMR systems are under development to handle the much greaterneeds of solving the full set of 3D Einstein equations. A large collaboration,begun by the NSF Black Hole Grand Challenge Alliance, has been developing asystem for distributing computing on large parallel machines, called DistributedAdapted Grid Hierarchies, or DAGH. DAGH was developed to provide MPIbasedparallelism for the kinds of computations needed for many PDE solvers,and it also provides a framework for parallel AMR. It is one of the majorcomputational science accomplishments to come out of the Alliance. Developedby Manish Parashar and Jim Browne, in collaboration with many subgroupswithin and without the Alliance, it is now being applied to many problemsin science and engineering. One can find information about DAGH online athttp://www.cs.utexas.edu/users/dagh/.At least two other 3D software environments for AMR have been developedfor relativity: one is called HLL, or Hierarchical Linked Lists, developed byWild and Schutz [140]; another, called BAM, was the first AMR application in3D relativity developed by Brügmann [60]. The HLL system has recently beenapplied to the test problem of the Zerilli equation (discussed above) describingperturbations of black holes [141]. This nearly 30 year old linear equation is stillproviding a powerful model for studying BH collisions, and it is also being usedas a model problem for 3D AMR. In this work, the 1D Zerilli equation is recastas a 3D equation in Cartesian coordinates, and evolved within the AMR systemprovided by HLL. Even though the 3D Zerilli equation is a single linear equation,it is quite demanding in terms of resolution requirements, and without AMR it isextremely difficult to resolve both the initial pulse of radiation, the blue shiftingof waves as they approach the horizon, and the scattering of radiation, includingthe normal modes, far from the hole.18.7 Recent applications and progress18.7.1 Evolving pure gravitational wavesWith the new formulations of the Einstein equations discussed above, we are nowable to study the nonlinear dynamics of pure gravitational waves with much morestability than ever before. This allows us to use numerical relativity to probegeneral relativity in highly nonlinear regime. Can one form a black hole infull 3D from pure gravitational waves? Does one see critical phenomena in full3D? These inherently nonlinear phenomena have been investigated in 1D and 2D