The Use of Richardson Extrapolation in One-Step Methods with Variable Step Size*

1. Introduction. One of the objections to the use of a one-step method to integrate a system of ordinary differential equations is that an estimate of the accumulated truncation error is difficult to make. If an attempt is made at appraising the truncation error, it is usually confined to an approximate evaluation of the local truncation error. A scheme for estimating the local truncation error, devised by Richardson [3], is based on the results of numerical integrations with steps h and h/2. The use of Richardson's extrapolation is well known (see, for example, [1, p. 81], [2, p. 238]). It is the purpose of this paper to show that it is possible to use the Richardson extrapolation procedure to form a useful estimate of the accumulated truncation error for a general one-step method even when the step size is allowed to vary. By using the estimate for the accumulated truncation error the accuracy of the numerical solution can be increased. Numerical examples to illustrate the estimation procedure are included.