  This work outlines a time-domain numerical integration technique for linear
hyperbolic partial differential equations sourced by distributions (Dirac
$\delta$-functions and their derivatives). Such problems arise when studying
binary black hole systems in the extreme mass ratio limit. We demonstrate that
such source terms may be converted to effective domain-wide sources when
discretized, and we introduce a class of time-steppers that directly account
for these discontinuities in time integration. Moreover, our time-steppers are
constructed to respect time reversal symmetry, a property that has been
connected to conservation of physical quantities like energy and momentum in
numerical simulations. To illustrate the utility of our method, we numerically
study a distributionally-sourced wave equation that shares many features with
the equations governing linear perturbations to black holes sourced by a point
mass.
