Doctoral Dissertations
Date of Award
8-1988
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Major
Mathematics
Major Professor
Steven M. Serbin
Abstract
In this dissertation a class of methods that numerically solve initial-value problems with second order ordinary differential equations of the form y" = f(x,y(x)) is investigated. Methods in this class are two step implicit Runge-Kutta methods with s internal stages that do not require an update of y'. There are many examples in the literature of methods which conform to our format.
Using a type of Nyström tree and a corresponding special type of Nyström series the order conditions for this method are developed. With this technique of putting order conditions in terms of trees, we obtain a set of simplifying conditions that serve as a framework for generating and analyzing higher order methods.
Our analysis affords the development of a two-parameter family of eighth-order methods. The issue of maximum obtainable order for unconditionally stable s stage methods is investigated for s = 1,2.
When implemented, these methods, in general, require at each step the solution of an algebraic equation of the form →Y= (M ⊗ Im)F(Y), Y ε n where M is an (s + 1)x(s - 1) matrix. To facilitate solving this equation we develop a method where M is lower triangular.
Recommended Citation
Cohen, Elizabeth Bruce, "Analysis of a class of two-step implicit Runge-Kutta schemes for second-order systems of ordinary differential equations. " PhD diss., University of Tennessee, 1988.
https://trace.tennessee.edu/utk_graddiss/11840
