Multi-Order Runge-Kutta Methods or how to numerically solve initial value problems of any order

Abstract

When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order initial value problems without such rewriting, however, allows for significantly greater accuracy. We therefore introduce a new generalization of Runge-Kutta methods, multi-order Runge-Kutta methods, designed to solve initial value problems of arbitrary order. We establish fundamental properties of these methods, including convergence, order of consistency, and linear stability. We also analyze the structure of the system satisfied by the approximations of a method, which enables us to provide a proper definition of explicit methods and gain a finer understanding of implicit methods.