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We revisit the problem of the emission of gravitational waves from a testmass orbiting and thus perturbing a Kerr black hole. The source term of theTeukolsky perturbation equation contains a Dirac delta function whichrepresents a point particle. We present a technique to effectively model thedelta function and its derivatives using as few as four points on a numericalgrid. The source term is then incorporated into a code that evolves theTeukolsky equation in the time domain as a (2+1) dimensional PDE. The waveformsand energy fluxes are extracted far from the black hole. Our comparisons withearlier work show an order of magnitude gain in performance (speed) andnumerical errors less than 1% for a large fraction of parameter space. As afirst application of this code, we analyze the effect of finite extractionradius on the energy fluxes. This paper is the first in a series whose goal isto develop adiabatic waveforms describing the inspiral of a small compact bodyinto a massive Kerr black hole.Comment: 21 pages, 6 figures, accepted by PRD. This version removes the  appendix; that content will be subsumed into future wor

Towards adiabatic waveforms for inspiral into Kerr black holes: A new model of the source for the time domain perturbation equation