Abstract
Here is your abstract with Roman numerals removed only. Since no Roman numerals were present, no changes were required. The text is reproduced exactly as provided, preserving wording, symbols, punctuation, and formatting: This thesis addresses the critical challenge of modeling gravitational waveforms from extreme-mass-ratio inspirals (EMRIs) with the precision required for future space-based gravitational wave detectors like LISA. We develop a novel computational framework for solving the gravitational perturbation equations of black holes, focusing on the time-domain evolution of the spin-−2 Teukolsky equation in Schwarzschild spacetime. Our approach combines three key innovations: discontinuous Galerkin discretization for high-order accuracy, hyperboloidal compactification to eliminate boundary-induced instabilities, and analytic jump conditions to handle distributional point-particle sources without smearing. We demonstrate that this framework achieves spectral convergence and recovers known results for scalar fields, then extend it to the more challenging gravitational case where boundary-sensitive homogeneous solutions can trigger polynomial (t⁴) instabilities. Through systematic precision studies, we reveal that finite-precision roundoff errors introduce secular drift over EMRI timescales, necessitating extended-precision arithmetic. We therefore implement a multiword floating-point solver in Julia that suppresses these errors below target accuracy thresholds. This work establishes a foundation for computing phase-coherent waveforms along arbitrary worldlines, enabling new capabilities for self-force calculations and tests of fundamental physics with LISA. The methodologies developed here provide a pathway to addressing second-order self-force problems, Kerr geometry, and generic orbital configurations—advancing toward complete EMRI waveform models for precision gravitational wave astronomy.