Continuous Explicit Runge-kutta Methods of Order 5

A continuous explicit Runge-Kutta (CERK) method provides a continuous approximation to an initial value problem. Such a method may be obtained by appending additional stages to a discrete method, or alternatively by solving the appropriate order conditions directly. Owren and Zennaro have shown for order 5 that the latter approach yields some CERK methods that require fewer derivative evaluations than methods obtained by appending stages. In contrast, continuous methods of order 6 that require the minimum number of stages can be obtained by appending additional stages to certain discrete methods. This article begins a study to understand why this occurs. By making no assumptions to simplify solution of the order conditions, the existence of other types of CERK methods of order 5 is established. While methods of the new families may not be as good for implementation as the Owren-Zennaro methods, the structure is expected to lead to a better understanding of how to construct families of methods of higher order.