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.
- The PNP problem
- Manas Vishal
- 0000-0003-3424-3505
- Scott E Field (Advisor) - University of Massachusetts Dartmouth, Department of MathematicsGaurav Khanna (Advisor) - University of Rhode IslandSigal Gottlieb (Committee Member) - University of Massachusetts Dartmouth, Department of MathematicsSarah Elizabeth Caudill (Committee Member) - University of Massachusetts Dartmouth, Department of Physics
- xiv, 160 pages
- illustrations (some color)
- List of figures -- List of tables -- Chapter 1. Introduction -- Motivation and context -- The geometric and numerical framework -- Thesis contributions and trajectory -- Organization of the thesis -- Chapter 2. A time-domain solver for the Scalar Teukolsky Equation -- Executive summary -- Introduction -- Evolution equations -- The discontinuous Galerkin Scheme -- Numerical experiments -- Summary and future work -- Chapter 3. Superconvergence of DG Methods for wave equations -- Introduction -- Preliminaries -- Superconvergence in discontinuous Galerkin methods -- Numerical experiments -- Summary -- Chapter 4. Time-domain evolution of the s = −2 Teukolsky equation in Schwarzschild -- Executive summary -- Motivation and background -- The s = −2 Teukolsky equation and its instabilities -- Mitigation via hyperboloidal compactification -- Alternative mitigation: Chandrasekhar transformation -- Numerical implementation and precision sensitivity -- Discussion and future directions -- Chapter 5. A Julia-based multiword-precision solver -- Executive summary -- Introduction -- Preliminaries -- Multiword-precision time-domain solver -- Implementing the solver in Julia -- Numerical experiments -- Summary -- Chapter 6. Conclusion.
- Includes bibliographical references (pages 148-160).
- University of Massachusetts Dartmouth
- Doctor of Philosophy (PHD)
- Engineering and Applied Science
- College of Engineering
- English
- Dissertation
- Copyright 2026 Manas Vishal
- https://doi.org/10.62791/20558
- 9914528161401301