Elements of a general theory of composite integration methods

摘要

This contribution is a survey of results on general linear methods. After an introduction into the concept of A-methods (which is a slight generalization of general linear methods) and its main results, RK-methods are presented in the A-method notation which offers the following advantages: (a) the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials; (b) they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the RK-method; (c) their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions; (d) the error accumulation within the stages of the method is completely understood. The proof of the main result on RK-methods (theorem 5) is simplified.