Numerical approaches to first- and second-order self-force calculations

Extreme-mass-ratio inspirals (EMRIs) - binary systems wherein a small compact object inspirals into a supermassive black hole - are considered to be important potential sources for the future space-based gravitational wave detector LISA. Inspirals of this nature are long-lived and are expected to possess a highly complex morphology with eccentric and precessing trajectories of the small compact object. Therefore systems of this nature are unique in the parameter space since they offer an unparalleled insight into the strong-field regime of general relativity through radiated gravitational waves (GWs). Achieving this scientific insight will require highly accurate waveform templates that remain phase accurate over these long inspirals. The characteristic small-ratio and stringent accuracy requirements necessitates the use of second-order self-force theory. Practical second-order self-force calculations are now emerging for quasi-circular EMRIs. In this dissertation I present novel computational approaches that look to extend current second-order self-force calculations to a more realistic scenarios of systems where the smaller companions are in eccentric inspirals and systems with a spinning primary. The techniques promise to model both EMRIs and intermediate-mass-ratio-inspirals (IMRIs). Furthermore, I present work on alternative hyperboloidal approaches to self-force calculations that offer to alleviate the computational burdens that current calculations face.