Analysis of a class of two-step implicit Runge-Kutta schemes for second-order systems of ordinary differential equations
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.
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