
Type of Document Dissertation Author Kovacs, Mihaly Author's Email Address kmisi@math.lsu.edu URN etd07082004143318 Title On Qualitative Properties and Convergence of TimeDiscretization Methods for Semigroups Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee
Advisor Name Title Frank Neubrander Committee Chair Istvan Farago Committee CoChair Augusto Nobile Committee Member HuiHsiung Kuo Committee Member Robert Perlis Committee Member Stephen Shipman Committee Member Ramachandran Vaidyanathan Dean's Representative Keywords
 stability of timediscretization methods
 positivity preservation
 normbound preservation
 contractivity preservation
 finite difference schemes
 HillePhillips functional calculus
 convexity preservation
 LaplaceStieltjes transform
 intermediate spaces
 interpolation of linear operators
 rational approximation of semigroups
 convergence of timediscretization methods
 Favard spaces
Date of Defense 20040708 Availability unrestricted Abstract In this dissertation we use functional calculus methods to investigate convergenceand qualitative properties of timediscretization methods for strongly continuous semigroups.
Stability, convergence, and preservation of contractivity (or normbound) of the
semigroup under timediscretization 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 HillePhillips (HP) functional calculus instead of the DunfordTaylor
functional calculus allows us to extend fundamental qualitative results concerning
timediscretization methods and simplify their proofs, including results on multistep schemes and
variable stepsizes. 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 HP functional calculus is one of the main mathematical tools throughout the dissertation, we present an elementary introduction to it based on the RiemannStieltjes integral. Aside from theoretical investigations, we show how our functional analytic methods can be used for computational purposes by applying the results to the onedimensional heat equation. The DunfordTaylor functional calculus is employed to obtain an estimate on the stability constant of the restricted denominator approximation method applied to the one dimensional, spacediscretized heat equation. Finally, we propose a second order timediscretization method for the spacediscrete heat equation that preserves contractivity in the maximum norm for all timesteps.
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