A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit

We obtain a fourth-order accurate numerical algorithm to integrate the Zerilli and Regge–Wheeler wave equations, describing perturbations of non-rotating black holes, with source terms due to an orbiting particle. Those source terms contain Dirac's delta and its first derivative. We also re-derive the source of the Zerilli and Regge–Wheeler equations for more convenient definitions of the waveforms that allow direct metric reconstruction (in the Regge–Wheeler gauge).