Summary: 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).