  We revisit the problem of the emission of gravitational waves from a test
mass orbiting and thus perturbing a Kerr black hole. The source term of the
Teukolsky perturbation equation contains a Dirac delta function which
represents a point particle. We present a technique to effectively model the
delta function and its derivatives using as few as four points on a numerical
grid. The source term is then incorporated into a code that evolves the
Teukolsky equation in the time domain as a (2+1) dimensional PDE. The waveforms
and energy fluxes are extracted far from the black hole. Our comparisons with
earlier work show an order of magnitude gain in performance (speed) and
numerical errors less than 1% for a large fraction of parameter space. As a
first application of this code, we analyze the effect of finite extraction
radius on the energy fluxes. This paper is the first in a series whose goal is
to develop adiabatic waveforms describing the inspiral of a small compact body
into a massive Kerr black hole.
