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Type of Document Dissertation Author Kovacs, Mihaly Author's Email Address kmisi@math.lsu.edu URN etd-07082004-143318 Title On Qualitative Properties and Convergence of Time-Discretization Methods for Semigroups Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee
Advisor Name Title Frank Neubrander Committee Chair Istvan Farago Committee Co-Chair Augusto Nobile Committee Member Hui-Hsiung Kuo Committee Member Robert Perlis Committee Member Stephen Shipman Committee Member Ramachandran Vaidyanathan Dean's Representative Keywords
- 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
Date of Defense 2004-07-08 Availability unrestricted Abstract In this dissertation we use functional calculus methods to investigate convergenceand 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 X 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.
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