Rapid evaluation of radiation boundary kernels for time-domain wave propagation on black holes: implementation and numerical tests

For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild black hole background, we consider the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the black hole. Such boundary conditions feature temporal integral convolution between each spherical harmonic mode of the wave field and a time-domain radiation kernel (TDRK). For each orbital angular integer l the associated TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK). Drawing upon theory and numerical methods developed in a previous article, 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. 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.