General linear methods have a long history, especially as regards
theoretical developments. For example the paper by Butcher [1]
presented a formulation for multistage-multivalue methods, which were
extended to include multiderivative terms by Hairer and Wanner
[5]. In contrast with this early work, there have been few
developments identifying new practical methods.
ARK methods (or Almost Runge Kutta methods) are a specific class of
general linear methods which have been developed to retain many of the
good properties of Runge-Kutta methods, while overcoming some of their
undesirable features. They retain the simple stability properties
of Runge-Kutta methods. However, they have the advantage that
error estimation and interpolation is cheap and easy to implement.
Amongst the large class of fourth order ARK methods, a method has been
identified which, when implemented in a careful way, behaves as though it
were fifth order. A discussion of this method will be presented,
including an explanation of what seem to be its desirable properties.