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 requi re an update of y'. There are many examples in the literature of metho ds which conform to our format. Using a type of Nystrom tree and a cor responding special type of Nystrom series, the order conditions for th is method are developed. With this technique of putting order conditio ns in terms of trees, we obtain a set of simplifying conditions that s erve as a framework for generating and analyzing higher order methods. Our analysis affords the development of a two-parameter family of eig hth-order methods. The issue of maximum obtainable order for unconditi onally stable s stage methods is investigated for s = 1, 2. When imple mented, these methods, in general, require at each step the solution o f an algebraic equation of the form Y = (M x I(m))F(Y), Y is-an-elemen t-of R(n), where M is an (s + 1) x (s + 1) matrix. To facilitate solvi ng this equation, we develop a method where M is lower triangular.