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 PDE an incarnation of the confluent Hewn 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 PDE. 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. We also consider a three-dimensional evolution based on a spectral code, one showing that the ROBC yield accurate results for the scenario of a wave packet striking the boundary at an angle. 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.