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

Abstract

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 εuniformly along the axis of imaginary Laplace frequency. Theoretically, εis 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.

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) 4147-4192

References in corpus (3)

• Boundary conditions in linearized harmonic gravity
• Absorbing boundary conditions for simulation of gravitational waves with spectral methods in spherical coordinates
• Radiation tails and boundary conditions for black hole evolutions

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