General linear methods

Abstract

General linear methods, as multistage multivalue methods, are the natural generalizations of linear multistep and Runge-Kutta methods. This survey contains a discussion of the traditional methods and a motivation for the general linear type of generalization. The new methods are introduced in terms of their formulation and the basic properties of consistency, stability and convergence. The order of general linear methods has to be looked at from a new point of view and it is shown how to use an algebraic structure (equivalent to B-series) to express conditions for a given order. Linear and non-linear stability for the new methods brings the theories for the classical methods into a comprehensive formulation and known results are outlined. Recently a number of subfamilies have been introduced and some of these are considered in detail. This applies in particular to methods with the property known as ‘inherent Runge-Kutta stability’. These seem to have prospects of yielding useful and efficient methods, and some progress towards their practical implementation is outlined. Finally, the relationship between stability functions and order of methods is discussed in a setting wide enough to include general linear methods as well as multiderivative methods, such as Obreshkov methods. The classical barriers due to Ehle, Daniel-Moore and Dahlquist (second barrier) all fit into a common pattern and these are explored in a general setting.