{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/116185"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/116185","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Time-symmetric integration of partial differential equations with applications to black hole physics","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_has_math":false,"creators":["O'Boyle, Michael F."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Physics","degree_department":null,"school":null,"contributors":["Gammie, Charles F","Markakis, Charalampos M","Ricker, Paul M","Witek, Helvi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08","date_published":"2022-08","updated_at":"2026-07-22T22:24:55Z","subjects":["Geometric integrators","numerical analysis","black hole perturbation theory","hyperbolic partial differential equations","hyperboloidal compactification","gravitational waves","neutron star equation of state"],"languages":["en","eng"],"rights":["Copyright 2022 Michael F O'Boyle"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/116185","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Gammie, Charles F","Markakis, Charalampos M","Ricker, Paul M","Witek, Helvi"]},{"key":"dc:creator","label":"Author","values":["O'Boyle, Michael F."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-08","2022-07-07"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Geometric integrators","numerical analysis","black hole perturbation theory","hyperbolic partial differential equations","hyperboloidal compactification","gravitational waves","neutron star equation of state"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2022 Michael F O'Boyle"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/116185"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Michael O'Boyle, accepted the attached license on 2022-07-06 at 10:39.","The student, Michael O'Boyle, submitted this Dissertation for approval on 2022-07-06 at 10:49.","This Dissertation was approved for publication on 2022-07-07 at 11:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18151 on 2022-11-15 at 17:38:12","The detection of gravitational waves from merging binary black holes has heralded a new era of astronomy: gravitational radiation has joined electromagnetic radiation and elementary particles as a channel through which we may observe the universe around us. This has created the need for accurate, robust models and calculations describing the predictions that general relativity and other theories make for a whole new class of exotic, complicated phenomena. One such phenomenon is the ``extreme mass ratio inspiral,'' wherein a star or stellar-mass black hole plunges into a supermassive black hole found in a galactic nucleus and emits gravitational waves. These events present the possibility to learn about the objects found under such conditions, which would reveal information about galaxy formation and stellar evolution and probe the validity of general relativity in these regions' strong gravitational fields. Such phenomena are incredibly complicated, with only the most trivial problems possessing closed-form solutions. The aim of this work is to present a series of tools to facilitate the numerical study of these phenomena. First, we will emphasize the need for numerical methods which accurately track the physics of the problem: i.e. respect the qualitative features of the evolution equations--symmetry under time-reversal and symplecticity--and accurately track conserved quantities of interest in gravitational wave astronomy: energy and angular momentum. Next, we will discuss the problem of representing a distributional source term, i.e. a point mass, on a discrete grid, and consider the effect this has on discrete operations like numerical differentiation and integration. After providing a brief overview of the relevant aspects of black hole theory, we will discuss how to extract gravitational waves at infinite distances from the black hole using numerical methods. Finally, we will demonstrate the methods' utility in working with problems of black hole theory, showing that conservative, geometric evolution is possible. Second, we will discuss a problem related to the astrophysics of neutron stars. A current goal of gravitational wave astronomy is the constraint of the neutron star thermodynamic equation of state, which will inform the fundamental theory of nuclear interactions. It is necessary to parametrize the equation of state to compare to observation. We present an extension of the popular piecewise polytrope formalism that allows for a continuous sound speed in the nuclear matter. We found that in addition to capturing the thermodynamics of many microscopic models, it more accurately predicts the astrophysically observable properties of neutron stars -- mass, radius, and tidal deformability -- when compared to the original model."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Time-symmetric integration of partial differential equations with applications to black hole physics"]}]}],"canonical_facts":{"dc:contributor":["Gammie, Charles F","Markakis, Charalampos M","Ricker, Paul M","Witek, Helvi"],"dc:creator":["O'Boyle, Michael F."],"dc:date":["2022-08","2022-07-07"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Michael O'Boyle, accepted the attached license on 2022-07-06 at 10:39.","The student, Michael O'Boyle, submitted this Dissertation for approval on 2022-07-06 at 10:49.","This Dissertation was approved for publication on 2022-07-07 at 11:04.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18151 on 2022-11-15 at 17:38:12","The detection of gravitational waves from merging binary black holes has heralded a new era of astronomy: gravitational radiation has joined electromagnetic radiation and elementary particles as a channel through which we may observe the universe around us. This has created the need for accurate, robust models and calculations describing the predictions that general relativity and other theories make for a whole new class of exotic, complicated phenomena. One such phenomenon is the ``extreme mass ratio inspiral,'' wherein a star or stellar-mass black hole plunges into a supermassive black hole found in a galactic nucleus and emits gravitational waves. These events present the possibility to learn about the objects found under such conditions, which would reveal information about galaxy formation and stellar evolution and probe the validity of general relativity in these regions' strong gravitational fields. Such phenomena are incredibly complicated, with only the most trivial problems possessing closed-form solutions. The aim of this work is to present a series of tools to facilitate the numerical study of these phenomena. First, we will emphasize the need for numerical methods which accurately track the physics of the problem: i.e. respect the qualitative features of the evolution equations--symmetry under time-reversal and symplecticity--and accurately track conserved quantities of interest in gravitational wave astronomy: energy and angular momentum. Next, we will discuss the problem of representing a distributional source term, i.e. a point mass, on a discrete grid, and consider the effect this has on discrete operations like numerical differentiation and integration. After providing a brief overview of the relevant aspects of black hole theory, we will discuss how to extract gravitational waves at infinite distances from the black hole using numerical methods. Finally, we will demonstrate the methods' utility in working with problems of black hole theory, showing that conservative, geometric evolution is possible. Second, we will discuss a problem related to the astrophysics of neutron stars. A current goal of gravitational wave astronomy is the constraint of the neutron star thermodynamic equation of state, which will inform the fundamental theory of nuclear interactions. It is necessary to parametrize the equation of state to compare to observation. We present an extension of the popular piecewise polytrope formalism that allows for a continuous sound speed in the nuclear matter. We found that in addition to capturing the thermodynamics of many microscopic models, it more accurately predicts the astrophysically observable properties of neutron stars -- mass, radius, and tidal deformability -- when compared to the original model."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/116185"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Michael F O'Boyle"],"dc:subject":["Geometric integrators","numerical analysis","black hole perturbation theory","hyperbolic partial differential equations","hyperboloidal compactification","gravitational waves","neutron star equation of state"],"dc:title":["Time-symmetric integration of partial differential equations with applications to black hole physics"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Physics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:55Z"}