Two-step processes by one-step methods of order 3 and of order 4

Abstract

and the initial condition y(χo) = jo, where /(#, y) is assumed to be a sufficiently smooth function. We are concerned with the case where the equation (1.1) is integrated numerically by one-step methods of order 3 and of order 4. It is well known that the one-step methods of order 3 such as Kutta method and those of order 4 such as Runge-Kutta method require three and four evaluations of the derivative respectively. It is also known that, if the same step-size is used twice in succession, an approximate value of the truncation error can be obtained by integrating again with the double step-size. This method of approximating the truncation error requires, per two steps of integration, eight and eleven evaluations of f(x, y) for any one-step method of order 3 and for that of order 4 respectively. In our previous paper C19], it has been shown that there exists a onestep formula of order 4 such that, after two steps of integration with the same step-size, only one additional evaluation of the derivative makes/it possible to approximate the truncation error. In that formula, however, four values of f(χ, y) evaluated in the first step of integration are not used explicitly in the second step. Thus there remains a possibility of reducing the number of evaluations of the derivative by utilizing all the values of the derivative computed already. In this paper, it is shown that there exist one-step integration formulas of order 3 and those of of order 4 such that approximate values zx and z2 of y(χo + h) and y(xo-\-2h) and an approximation to their truncation errors can be obtained with five and seven evaluations of f(x, y) respectively. Finally two numerical examples are presented.