ANALYSIS OF A CLASS OF MULTISTAGE, MULTISTEP RUNGE-KUTTA METHODS

In this paper, a class of methods that numerically solve initial-value problems for 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 Nystrom tree and a corresponding special type of Nystrom 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 x I(m))F(Y), Y is-an-element-of R(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.