Rapid Evaluation of Radiation Boundary Kernels for Time-domain Wave Propagation on Blackholes

For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild blackhole background, we describe the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the ROBC is based on Laplace and spherical-harmonic transformation of the Regge-Wheeler equation, the PDE governing the wave propagation, with the resulting radial ODE an incarnation of the confluent Heun equation. For a given angular index l the ROBC feature integral convolution between a time-domain radiation boundary kernel (TDRK) and each of the corresponding 2l+1 spherical-harmonic modes of the radiating wave. The TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ODE. We numerically implement the ROBC via a rapid algorithm involving approximation of the FDRK by a rational function. Such an approximation is tailored to have relative error \epsilon uniformly along the axis of imaginary Laplace frequency. Theoretically, \epsilon is also a long-time bound on the relative convolution error. Via study of one-dimensional radial evolutions, we demonstrate that the ROBC capture the phenomena of quasinormal ringing and decay tails. Moreover, carrying out a numerical experiment in which a wave packet strikes the boundary at an angle, we find that the ROBC yield accurate results in a three-dimensional setting. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom.Comment: AMS article, 105 pages, 45 figures. Version 3 has more minor corrections as well as extra commentary added in response to reactions by referees. Commentary added which compares and contrasts this work with work of Leaver and work of Andersson. For publication, article has been cut in two and appears as two separate articles in J. Comp. Phys. 199 (2004) 376-422 and Class. Quantum Grav. 21 (2004) 4147-419

IMEX evolution of scalar fields on curved backgrounds

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.Comment: 28 pages, uses revtex4. Revised in response to referee's report. One numerical experiment added which incorporates perturbed initial data and adaptive time-steppin

Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains

We describe a multidomain spectral-tau method for solving the three-dimensional helically reduced wave equation on the type of two-center domain that arises when modeling compact binary objects in astrophysical applications. A global two-center domain may arise as the union of Cartesian blocks, cylindrical shells, and inner and outer spherical shells. For each such subdomain, our key objective is to realize certain (differential and multiplication) physical-space operators as matrices acting on the corresponding set of modal coefficients. We then achieve sparse realizations through the integration “preconditioning” of Coutsias, Hagstrom, Hesthaven, and Torres. Since ours is the first three-dimensional multidomain implementation of the technique, we focus on the issue of convergence for the global solver, here the alternating Schwarz method accelerated by GMRES. Our methods may prove relevant for numerical solution of other mixed-type or elliptic problems, and in particular for the generation of initial data in general relativity

Persistent junk solutions in time-domain modeling of extreme mass ratio binaries

In the context of metric perturbation theory for non-spinning black holes, extreme mass ratio binary (EMRB) systems are described by distributionally forced master wave equations. Numerical solution of a master wave equation as an initial boundary value problem requires initial data. However, because the correct initial data for generic-orbit systems is unknown, specification of trivial initial data is a common choice, despite being inconsistent and resulting in a solution which is initially discontinuous in time. As is well known, this choice leads to a "burst" of junk radiation which eventually propagates off the computational domain. We observe another unintended consequence of trivial initial data: development of a persistent spurious solution, here referred to as the Jost junk solution, which contaminates the physical solution for long times. This work studies the influence of both types of junk on metric perturbations, waveforms, and self-force measurements, and it demonstrates that smooth modified source terms mollify the Jost solution and reduce junk radiation. Our concluding section discusses the applicability of these observations to other numerical schemes and techniques used to solve distributionally forced master wave equations.Comment: Uses revtex4, 16 pages, 9 figures, 3 tables. Document reformatted and modified based on referee's report. Commentary added which addresses the possible presence of persistent junk solutions in other approaches for solving master wave equation

Genome-wide detection of segmental duplications and potential assembly errors in the human genome sequence

BACKGROUND: Previous studies have suggested that recent segmental duplications, which are often involved in chromosome rearrangements underlying genomic disease, account for some 5% of the human genome. We have developed rapid computational heuristics based on BLAST analysis to detect segmental duplications, as well as regions containing potential sequence misassignments in the human genome assemblies. RESULTS: Our analysis of the June 2002 public human genome assembly revealed that 107.4 of 3,043.1 megabases (Mb) (3.53%) of sequence contained segmental duplications, each with size equal or more than 5 kb and 90% identity. We have also detected that 38.9 Mb (1.28%) of sequence within this assembly is likely to be involved in sequence misassignment errors. Furthermore, we have identified a significant subset (199,965 of 2,327,473 or 8.6%) of single-nucleotide polymorphisms (SNPs) in the public databases that are not true SNPs but are potential paralogous sequence variants. CONCLUSION: Using two distinct computational approaches, we have identified most of the sequences in the human genome that have undergone recent segmental duplications. Near-identical segmental duplications present a major challenge to the completion of the human genome sequence. Potential sequence misassignments detected in this study would require additional efforts to resolve

Rapid evaluation of radiation boundary kernels for time-domain wave propagation on blackholes: theory and numerical methods

For scalar, electromagnetic, or gravitational wave propagation on a background Schwarzschild blackhole, we describe the exact nonlocal radiation outer boundary conditions (robc) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the robc is based on Laplace and spherical–harmonic transformation of the Regge–Wheeler equation, the pde governing the wave propagation, with the resulting radial ode an incarnation of the confluent Heun equation. For a given angular integer l the robc feature integral convolution between a time–domain radiation boundary kernel (tdrk) and each of the corresponding 2l+1 spherical–harmonic modes of the radiating wave field. The tdrk is the inverse Laplace transform of a frequency–domain radiation kernel (fdrk) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ode. We numerically implement the robc via a rapid algorithm involving approximation of the fdrk by a rational function. Such an approximation is tailored to have relative error ε uniformly along the axis of imaginary Laplace frequency. Theoretically, ε is also a long–time bound on the relative convolution error. Via study of one–dimensional radial evolutions, we demonstrate that the robc capture the phenomena of quasinormal ringing and decay tails. Moreover, carrying out a numerical experiment in which a wave packet strikes the boundary at an angle, we find that the robc yield accurate results in a three–dimensional setting. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (agh), save for one key difference. Whereas agh had the usual armamentarium of analytical results (asymptotics, order recursion relations, bispectrality) for Bessel functions at their disposal, what we need to know about Heun functions must be gathered numerically as relatively less is known about them. Therefore, unlike agh, we are unable to offer an asymptotic analysis of our rapid implementation

New variables, the gravitational action, and boosted quasilocal stress-energy-momentum

This paper presents a complete set of quasilocal densities which describe the stress-energy-momentum content of the gravitational field and which are built with Ashtekar variables. The densities are defined on a two-surface $B$ which bounds a generic spacelike hypersurface $\Sigma$ of spacetime. The method used to derive the set of quasilocal densities is a Hamilton-Jacobi analysis of a suitable covariant action principle for the Ashtekar variables. As such, the theory presented here is an Ashtekar-variable reformulation of the metric theory of quasilocal stress-energy-momentum originally due to Brown and York. This work also investigates how the quasilocal densities behave under generalized boosts, i. e. switches of the $\Sigma$ slice spanning $B$. It is shown that under such boosts the densities behave in a manner which is similar to the simple boost law for energy-momentum four-vectors in special relativity. The developed formalism is used to obtain a collection of two-surface or boost invariants. With these invariants, one may ``build" several different mass definitions in general relativity, such as the Hawking expression. Also discussed in detail in this paper is the canonical action principle as applied to bounded spacetime regions with ``sharp corners."Comment: Revtex, 41 Pages, 4 figures added. Final version has been revised and improved quite a bit. To appear in Classical and Quantum Gravit