Doctor of Philosophy (PhD)
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 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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Kovacs, Mihaly, "On qualitative properties and convergence of time-discretization methods for semigroups" (2004). LSU Doctoral Dissertations. 1244.