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# Algebraic structures for the analysis of numerical methods

Abstract (Summary)
Restricted Item. Print thesis available in the University of Auckland Library or available through Inter-Library Loan. The starting point in this thesis is a new formulation of the composition rule for the algebraic structure associated with Runge-Kutta methods. This new formulation enables us to find a new way of generating the composition rule recursively. It turns out that this new composition rule is more efficient. This new formula also motivates the study of the tree space, forest space, dual forest space. Following on from the new insight of the study of the linear structure of the set of rooted trees, we also consider several consequences and refinements, including: Using linear combinations of trees together with the order condition derived from this linear combination by the Picard integral. This simplifies the calculations in solving the order conditions for integration methods. A formal proof of the basis for an integration method satisfying C and D conditions. A proof that the set of elementary weight functions satisfying D conditions forms a group. Using a set of Runge-Kutta methods to approach the Picard integral and regarding the Picard integral as a limit of Runge-Kutta methods. Comparisons and relationships between Butcher's approach to the Taylor series expansion and the B-series (a reformulation of Butcher's approach by Hairer and Wanner) are presented. Originally derived by analysing Runge-Kutta methods, the algebraic structure is applicable to other kinds of methods and differential systems. We believe that the insight of this study is helpful in developing simpler structures for other differential systems. Demonstrating the applications to the NystrÃ¶m system and Almost Runge-Kutta methods, we want to lay a stepping stone to these applications. We are also concerned with more modern applications of the algebraic structure. By analysing the principal local truncation errors for two methods with the same order, we are able to get a composition method of one order higher. A new error estimator, an "m-step zero approximation", is proposed. Several applications for this error estimator are discussed. A changing stepsize scheme for effective order methods is proposed and is successfully implemented. We use the following diagram to show the motivations and the main streams of this thesis.
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