{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/116225"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/116225","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Discontinuous differential equations","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":["Wojtalewicz, Nikolas"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hirani, Anil","Wan, Andy","DeVille, Lee","Laugesen, Richard"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08","date_published":"2022-08","updated_at":"2026-07-22T22:24:55Z","subjects":["Numerical Analysis","Ordinary Differential Equations","Partial Differential Equations","Numerical Methods"],"languages":["en","eng"],"rights":["Copyright 2022 Nikolas Wojtalewicz"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/116225","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hirani, Anil","Wan, Andy","DeVille, Lee","Laugesen, Richard"]},{"key":"dc:creator","label":"Author","values":["Wojtalewicz, Nikolas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-08","2022-07-14"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"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":["Numerical Analysis","Ordinary Differential Equations","Partial Differential Equations","Numerical Methods"]}]},{"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 Nikolas Wojtalewicz"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/116225"]}]},{"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, Nikolas Wojtalewicz, accepted the attached license on 2022-07-12 at 20:56.","The student, Nikolas Wojtalewicz, submitted this Dissertation for approval on 2022-07-12 at 21:06.","This Dissertation was approved for publication on 2022-07-14 at 12:57.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18272 on 2022-11-15 at 18:20:48","This thesis covers numerical methods for discontinuous differential equations. In the context of ordinary differential equations, we introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order and conserved quantities across the interface. In the context of partial differential equations we introduce a numerical method based on discrete exterior calculus for the phase field equation. We prove the phase field variable remains bounded and satisfies mass conservation. Further, our method works on embedded two-dimensional Delaunay meshes in $\\mathbb{R}^3$. Numerical examples on an embedded cylinder in $\\mathbb{R}^3$ demonstrate both boundedness and mass conservation up to machine precision."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Discontinuous differential equations"]}]}],"canonical_facts":{"dc:contributor":["Hirani, Anil","Wan, Andy","DeVille, Lee","Laugesen, Richard"],"dc:creator":["Wojtalewicz, Nikolas"],"dc:date":["2022-08","2022-07-14"],"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, Nikolas Wojtalewicz, accepted the attached license on 2022-07-12 at 20:56.","The student, Nikolas Wojtalewicz, submitted this Dissertation for approval on 2022-07-12 at 21:06.","This Dissertation was approved for publication on 2022-07-14 at 12:57.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18272 on 2022-11-15 at 18:20:48","This thesis covers numerical methods for discontinuous differential equations. In the context of ordinary differential equations, we introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order and conserved quantities across the interface. In the context of partial differential equations we introduce a numerical method based on discrete exterior calculus for the phase field equation. We prove the phase field variable remains bounded and satisfies mass conservation. Further, our method works on embedded two-dimensional Delaunay meshes in $\\mathbb{R}^3$. Numerical examples on an embedded cylinder in $\\mathbb{R}^3$ demonstrate both boundedness and mass conservation up to machine precision."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/116225"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Nikolas Wojtalewicz"],"dc:subject":["Numerical Analysis","Ordinary Differential Equations","Partial Differential Equations","Numerical Methods"],"dc:title":["Discontinuous differential equations"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"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"}