ABSTRACT
In the last chapter we asked how well the numerical methods approximate the solution of a differential equation over a single step. In this chapter we ask how well they approximate the solution over an interval [a, b]. In the classical theory the step size is a constant h and numerical methods are investigated for “small” h by letting h tend to 0. For one-step methods it turns out to be as easy to analyze variable step size as constant. Adams methods and others of similar form can be analyzed in a way rather like that used for one-step methods. In particular, it is possible to account for variation of the step size. Unfortunately, the important BDFs do not have this form and if we are to study them in a relatively simple manner, we have to make the classical assumption that the step size is constant. There are other reasons for taking up the special case of constant step size. For one, it is possible to understand better the behavior of LMMs and predictor-corrector methods. This will show us why some methods are of no practical value despite an origin and appearance that are quite similar to those of methods important in practice. For another, some of the tools developed for this analysis are needed for the study of the stability of methods when the step size is not “small” which is taken up in the next chapter. Variation of the step size is of great practical importance, so it is necessary to consider how to extend the classical approach of constant step size to accommodate it. The discrete variable methods we study produce approximate solutions on a mesh in the interval [a, b]. Taylor series methods produce approximate solutions everywhere and the same is true of the Adams methods and the BDFs. We shall see that the approximations between mesh points have the same order of accuracy as the approximations at the mesh points themselves.
