Summary:
This PhD-thesis contains an introduction and six research papers
sorted chronologically, of which the first four are accepted for
publication. The introduction aims at giving a very brief summary of
the background theory needed for the following papers. Also, some
motivation of the issues addressed by the papers is given. Paper I
discusses algebraic structures of ordered rooted trees and their
applications to Lie group integrators. Results from Hopf algebra
theory on elementary differentials for Lie group integrators are used,
and applications to order analysis and backward error analysis are
given. Paper II, III, IV, and V are primarily on exponential
integrators, a class of numerical schemes tailored the solution of
stiff systems of systems of ordinary differential equations. Paper II
discusses classical order analysis and gives some theoretical results
on the form of the integrators, applicable for the construction of
high order exponential integrators. Paper III is on an implementation
of exponential integrators on computers, and source code, available
electronically, accompanies the paper. Paper IV includes an
analytical and numerical study of the performance of two classes of
exponential integrators on the nonlinear Schrödinger equation.
Paper V is a numerical study of behaviour over long
integration invervals on the nonlinear Schrödinger equation, using
nonlinear spectral theory for determining validity of the numerical
solution and thereby jugdging the numerical integrators. At last, in
Paper VI, properties of a class of exponential like functions,
essential in exponential integrators, are derived, using an approach
based on Lie group theory.
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