B-Series and Multivalue Methods

Abstract

Multivalue methods have a long history in the form of linear multistep methods. In this chapter, an amalgam of multivalue and multistage (Runge–Kutta) methods is considered as a family of method, in its own right and given the name “general linear methods”. After a review of linear multistep methods, the prototypical multivalue methods, it is shown by example that new methods flow from these by allowing multiple vector field calculations. Similarly, Runge–Kutta methods, the prototypical one-step methods, are also simply examples of known, and not so well known, multistage multivalue methods. The insight provided by this wide range of example methods underlines the use of the natural and highly flexible general linear formulation. The fundamental questions of consistency, order, and convergence, take on a natural and straightforward meaning in the general context. The theory of order for these methods is an important application of B-series analysis. This is closely related to the existence of the underlying one-step method and the theory of invariant subspaces. Throughout the chapter new methods are introduced and analysed.