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Gravitational wave signals from extreme mass ratio inspirals are a key targetfor space-based gravitational wave detectors. These systems are typicallymodeled as a distributionally-forced Teukolsky equation, where the smallerblack hole is treated as a Dirac delta distribution. Time-domain solvers oftenuse regularization approaches that approximate the Dirac distribution thatoften introduce small length scales and are a source of systematic error,especially near the smaller black hole. We describe a multi-domaindiscontinuous Galerkin method for solving the distributionally-forced Teukolskyequation that describes scalar fields evolving on a Kerr spacetime. To handlethe Dirac delta, we expand the solution in spherical harmonics and recast thesourced Teukolsky equation as a first-order, one-dimensional symmetrichyperbolic system. This allows us to derive the method's numerical flux tocorrectly account for the Dirac delta. As a result, our method achieves globalspectral accuracy even at the source's location. To connect the near field tofuture null infinity, we use the hyperboloidal layer method, allowing us tosupply outer boundary conditions and providing direct access to the far-fieldwaveform. We document several numerical experiments where we test our method,including convergence tests against exact solutions, energy luminosities forcircular orbits, the scheme's superconvergence properties at future nullinfinity, and the late-time tail behavior of the scalar field. We also comparetwo systems that arise from different choices of the first-order reductionvariables, finding that certain choices are numerically problematic inpractice. The methods developed here may be beneficial when computinggravitational self-force effects, where the regularization procedure has beendeveloped for the spherical harmonic modes and high accuracy is needed at theDirac delta's location.

Towards exponentially-convergent simulations of extreme-mass-ratio  inspirals: A time-domain solver for the scalar Teukolsky equation with  singular source terms