{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-2243"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-2243","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"On qualitative properties and convergence of time-discretization methods for semigroups","abstract":"In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We also generalize a basic result on the rate of convergence of rational approximation schemes for semigroups. We obtain convergence results on a continuum of intermediate spaces between the Banach space <i>X</i> and the domain of a certain power of the generator of the semigroup. The sharpness of these results is also discussed. Since the H-P functional calculus is one of the main mathematical tools throughout the dissertation, we present an elementary introduction to it based on the Riemann-Stieltjes integral. Aside from theoretical investigations, we show how our functional analytic methods can be used for computational purposes by applying the results to the one-dimensional heat equation. The Dunford-Taylor functional calculus is employed to obtain an estimate on the stability constant of the restricted denominator approximation method applied to the one dimensional, space-discretized heat equation. Finally, we propose a second order time-discretization method for the space-discrete heat equation that preserves contractivity in the maximum norm for all time-steps.","abstract_html":"In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We also generalize a basic result on the rate of convergence of rational approximation schemes for semigroups. We obtain convergence results on a continuum of intermediate spaces between the Banach space &lt;i&gt;X&lt;/i&gt; and the domain of a certain power of the generator of the semigroup. The sharpness of these results is also discussed. Since the H-P functional calculus is one of the main mathematical tools throughout the dissertation, we present an elementary introduction to it based on the Riemann-Stieltjes integral. Aside from theoretical investigations, we show how our functional analytic methods can be used for computational purposes by applying the results to the one-dimensional heat equation. The Dunford-Taylor functional calculus is employed to obtain an estimate on the stability constant of the restricted denominator approximation method applied to the one dimensional, space-discretized heat equation. Finally, we propose a second order time-discretization method for the space-discrete heat equation that preserves contractivity in the maximum norm for all time-steps.","abstract_has_math":false,"creators":["Kovacs, Mihaly"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004-01-01T08:00:00Z","date_published":"2004-01-01T08:00:00Z","updated_at":"2026-07-24T02:59:15Z","subjects":["stability of time-discretization methods","positivity preservation","norm-bound preservation","contractivity preservation","finite difference schemes","Hille-Phillips functional calculus","convexity preservation","Laplace-Stieltjes transform","intermediate spaces","interpolation of linear operators","rational approximation of semigroups","convergence of time-discretization methods","Favard spaces"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07082004-143318","https://repository.lsu.edu/gradschool_dissertations/1244"],"render_values":[{"text":"etd-07082004-143318","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/1244","href":"https://repository.lsu.edu/gradschool_dissertations/1244","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.1244","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Kovacs, Mihaly"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2004-07-08"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:11:29Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["stability of time-discretization methods","positivity preservation","norm-bound preservation","contractivity preservation","finite difference schemes","Hille-Phillips functional calculus","convexity preservation","Laplace-Stieltjes transform","intermediate spaces","interpolation of linear operators","rational approximation of semigroups","convergence of time-discretization methods","Favard spaces"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07082004-143318","10.31390/gradschool_dissertations.1244","https://repository.lsu.edu/gradschool_dissertations/1244"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We also generalize a basic result on the rate of convergence of rational approximation schemes for semigroups. We obtain convergence results on a continuum of intermediate spaces between the Banach space <i>X</i> and the domain of a certain power of the generator of the semigroup. The sharpness of these results is also discussed. Since the H-P functional calculus is one of the main mathematical tools throughout the dissertation, we present an elementary introduction to it based on the Riemann-Stieltjes integral. Aside from theoretical investigations, we show how our functional analytic methods can be used for computational purposes by applying the results to the one-dimensional heat equation. The Dunford-Taylor functional calculus is employed to obtain an estimate on the stability constant of the restricted denominator approximation method applied to the one dimensional, space-discretized heat equation. Finally, we propose a second order time-discretization method for the space-discrete heat equation that preserves contractivity in the maximum norm for all time-steps."]},{"key":"dc:title","label":"Title","values":["On qualitative properties and convergence of time-discretization methods for semigroups"]}]}],"canonical_facts":{"dc:creator":["Kovacs, Mihaly"],"dc:date":["2004-07-08"],"dc:date.available":["2022-05-12T23:11:29Z"],"dc:description.abstract":["In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We also generalize a basic result on the rate of convergence of rational approximation schemes for semigroups. We obtain convergence results on a continuum of intermediate spaces between the Banach space <i>X</i> and the domain of a certain power of the generator of the semigroup. The sharpness of these results is also discussed. Since the H-P functional calculus is one of the main mathematical tools throughout the dissertation, we present an elementary introduction to it based on the Riemann-Stieltjes integral. Aside from theoretical investigations, we show how our functional analytic methods can be used for computational purposes by applying the results to the one-dimensional heat equation. The Dunford-Taylor functional calculus is employed to obtain an estimate on the stability constant of the restricted denominator approximation method applied to the one dimensional, space-discretized heat equation. Finally, we propose a second order time-discretization method for the space-discrete heat equation that preserves contractivity in the maximum norm for all time-steps."],"dc:identifier":["etd-07082004-143318","10.31390/gradschool_dissertations.1244","https://repository.lsu.edu/gradschool_dissertations/1244"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["stability of time-discretization methods","positivity preservation","norm-bound preservation","contractivity preservation","finite difference schemes","Hille-Phillips functional calculus","convexity preservation","Laplace-Stieltjes transform","intermediate spaces","interpolation of linear operators","rational approximation of semigroups","convergence of time-discretization methods","Favard spaces"],"dc:title":["On qualitative properties and convergence of time-discretization methods for semigroups"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:59:15Z"}