Long simulations of extreme mass ratio inspirals by solving the Teukolsky equation with singular source terms

The Laser Interferometer Space Antenna (LISA), a future space-borne gravitational wave detector, is expected to detect extreme mass ratio inspirals (EMRIs). These are black hole mergers with a mass ratio greater than 100,000. EMRI science requires precise waveform models. Due to the extreme mass ratios involved, waveforms from these systems can be accurately found by numerically solving the Teukolsky equation sourced by a distributional source term. Solving the time-domain Teukolsky equation for spin-weight s=-2 in Boyer-Lindquist coordinates is complicated by (i) source terms that feature at least second derivatives of a Dirac delta function, and (ii) the unbounded slow growth of unphysical modes that contaminate long-duration simulations. We describe the development of a discontinuous Galerkin method, which exactly treats the distributional source as a modification to the numerical flux. We apply our method to long-time simulations and discuss approaches for dealing with the slow-growing, spurious modes caused by outer boundary conditions, by using techniques such as hyperboloidal layer methods, and suitable first-order reductions of the Teukolsky equation.