The tree and forest spaces with applications to initial-value problem methods
Tree and forest spaces, which are at the heart of the theory of Runge–Kutta methods, are formulated recursively, and it is shown that the forest space is an algebra. To obtain order conditions in a systematic manner, Banach algebras are introduced to generate both the elementary weights for a general Runge–Kutta method and the corresponding quantities based on the Picard integral. To connect these two concepts, the Picard integral is written as the limiting case of an s-stage Runge–Kutta method, equivalent to s steps of the Euler method, as s tends to infinity. This approach makes it possible to make direct use of the tree space without going over to the dual space. By choosing linear combinations of trees, appropriate to a particular application, it is shown how to obtain alternative ways of writing the order conditions. This leads to a simpler and more direct derivation of particular methods.