  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.
