Technological advancements have allowed computing power to generate high resolution model s. As a result, greater stiffness has been introduced into systems of ordinary differential equations (ODEs) that arise from spatial discreti zation of partial differential equations (PDEs). The components of the solutions to these systems are coupled and changing at widely varying rates, which present problems for time-stepping methods. Krylov Subspace Spectral methods, developed by Dr. James Lambers, bridge the gap between explicit and implicit methods for stiff problems by computing each Fouier coefficient from an individualized approximation of the solution operator. KSS methods demonstrate a high order of accuracy, but their efficiency needs to be improved. We will carry out an asymptotic study to determine how these approximations behave at high frequencies to develop a formula to reduce the computation of each node while still achieving a high level of accuracy. Our numerical results will reveal that our method does prove to increase the efficiency as well as the accuracy of KSS methods.
","abstract_html":"<p>Technological advancements have allowed computing power to generate high resolution model s. As a result, greater stiffness has been introduced into systems of ordinary differential equations (ODEs) that arise from spatial discreti zation of partial differential equations (PDEs). The components of the solutions to these systems are coupled and changing at widely varying rates, which present problems for time-stepping methods. Krylov Subspace Spectral methods, developed by Dr. James Lambers, bridge the gap between explicit and implicit methods for stiff problems by computing each Fouier coefficient from an individualized approximation of the solution operator. KSS methods demonstrate a high order of accuracy, but their efficiency needs to be improved. We will carry out an asymptotic study to determine how these approximations behave at high frequencies to develop a formula to reduce the computation of each node while still achieving a high level of accuracy. Our numerical results will reveal that our method does prove to increase the efficiency as well as the accuracy of KSS methods.</p>","abstract_has_math":false,"creators":["Palchak, Elisabeth Marie"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["James Lambers","Joseph Kalibal","Haiyan Tian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-05-01T07:00:00Z","date_published":"2012-05-01T07:00:00Z","updated_at":"2026-07-24T05:45:06Z","subjects":["Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/510","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["James Lambers","Joseph Kalibal","Haiyan Tian"]},{"key":"dc:creator","label":"Author","values":["Palchak, Elisabeth Marie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2018-11-15T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/510"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Technological advancements have allowed computing power to generate high resolution model s. As a result, greater stiffness has been introduced into systems of ordinary differential equations (ODEs) that arise from spatial discreti zation of partial differential equations (PDEs). The components of the solutions to these systems are coupled and changing at widely varying rates, which present problems for time-stepping methods. Krylov Subspace Spectral methods, developed by Dr. James Lambers, bridge the gap between explicit and implicit methods for stiff problems by computing each Fouier coefficient from an individualized approximation of the solution operator. KSS methods demonstrate a high order of accuracy, but their efficiency needs to be improved. We will carry out an asymptotic study to determine how these approximations behave at high frequencies to develop a formula to reduce the computation of each node while still achieving a high level of accuracy. Our numerical results will reveal that our method does prove to increase the efficiency as well as the accuracy of KSS methods.
"]},{"key":"dc:title","label":"Title","values":["Rapid Approximation of Bilinear Forms Involving Matrix Functions Through Asymptotic Analysis of Gaussian Node Placement"]}]}],"canonical_facts":{"dc:contributor":["James Lambers","Joseph Kalibal","Haiyan Tian"],"dc:creator":["Palchak, Elisabeth Marie"],"dc:date.available":["2018-11-15T08:00:00Z"],"dc:description.abstract":["Technological advancements have allowed computing power to generate high resolution model s. As a result, greater stiffness has been introduced into systems of ordinary differential equations (ODEs) that arise from spatial discreti zation of partial differential equations (PDEs). The components of the solutions to these systems are coupled and changing at widely varying rates, which present problems for time-stepping methods. Krylov Subspace Spectral methods, developed by Dr. James Lambers, bridge the gap between explicit and implicit methods for stiff problems by computing each Fouier coefficient from an individualized approximation of the solution operator. KSS methods demonstrate a high order of accuracy, but their efficiency needs to be improved. We will carry out an asymptotic study to determine how these approximations behave at high frequencies to develop a formula to reduce the computation of each node while still achieving a high level of accuracy. Our numerical results will reveal that our method does prove to increase the efficiency as well as the accuracy of KSS methods.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/510"],"dc:subject":["Mathematics","Physical Sciences and Mathematics"],"dc:title":["Rapid Approximation of Bilinear Forms Involving Matrix Functions Through Asymptotic Analysis of Gaussian Node Placement"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:45:06Z"}