Time-domain framework for the Teukolsky equation with a particle source using comoving hyperboloidal coordinates

We present a scheme and implementation code for time-domain integration of the Teukolsky equation in 1+1 dimensions with a point-particle source, based on comoving, spatially compactified hyperboloidal coordinates. We demonstrate that the scheme evades the problem of nonphysical growing modes that plague some numerical evolution schemes without compactification. Our use of comoving coordinates greatly simplifies the application of jump conditions on the particle's worldline. We develop our method and test its performance for a scalar field on a Schwarzschild background, first for a circular geodesic orbit source and then for a scattering geodesic orbit. We then present a test implementation of the method for the $s = -2$ Teukolsky equation, illustrating its long-term stability and absence of growing-mode behavior through comparison with similar results using noncompactified characteristic coordinates. Our method paves the way to calculations of the full gravitational self-force in extreme-mass-ratio scattering.