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 obtain ed by appending additional stages to a discrete method, or alternative ly by solving the appropriate order conditions directly. Owren and Zen naro have shown for order 5 that the latter approach yields some CERK methods that require fewer derivative evaluations than methods obtaine d by appending stages. In contrast, continuous methods of order 6 that require the minimum number of stages can be obtained by appending add itional stages to certain discrete methods. This article begins a stud y 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, th e structure is expected to lead to a better understanding of how to co nstruct families of methods of higher order.