Semiexplicit A -Stable Runge-Kutta Methods

An s — 1 stage semiexplicit Runge-Kutta method is represented by an i X i real lower triangular matrix where the number of implicit stages is given by the number of nonzero diagonal elements. It is shown that the maximum order attainable is i when s 4; but when i = 6, an .4-stable method of order 5 is obtained. This method has five nonzero diagonal elements, and these elements are equal. Finally, a six stage bi- stable method of order six is given. Again, this method has five nonzero (and equal) di- agonal elements. 1. Introduction. Consider an initial value problem, for a system of ordinary dif- ferential equations, of the form x = f(x), x(t0) = x0. An s - 1 stage, semiexplicit, Runge-Kutta method computes a sequence of approximations