Initial Value Problems for ODEs: Multistep Methods

Abstract

We saw in Chap. 5 that (explicit) one–step methods are increasingly difficult to construct as one upgrades the order requirement. This is no longer true for multistep methods, where an increase in order is straightforward but comes with a price: a potential danger of instability. In addition, there are other complications such as the need for an initialization procedure and considerably more complicated procedures for changing the grid length. Yet, in terms of work involved, multistep methods are still among the most attractive methods.We discuss them along lines similar to one– step methods, beginning with a local description and examples and proceeding to the global description and problems of stiffness. By the very nature of multistep methods, the discussion of stability is now more extensive.