Approximate computation of errors in numerical integration of ordinary differential equations by one-step methods

Abstract

where f(x, y) is assumed to be a sufficiently smooth function. In numerous papers [1 — 16], various methods are obtained for bounding or approximating the errors in numerical integration of (1.1) by one-step methods with the aids of the functions that bound or approximate the function fy (#, y), the truncation error and so on. To avoid the use of such functions for practical purposes, in this paper, n steps of integration with a fixed step-size are considered as one step and a simple method is obtained for approximating the errors without computing explicitly any function other than f(x, y). The method is illustrated by two numerical examples. Since usually the step-size is not changed so often and the estimate of the error is not always necessary for each step of integration, it will not be a serious restriction to fix the step-size for the n steps of integration, and this method may be used as an integration method with a check on the accuracy of the numerical solution.