Title page for ETD etd-07082004-143318


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 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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