DIFFERENTIABLE INTERPOLANTS FOR HIGH-ORDER RUNGE-KUTTA METHODS

For a particular family of pairs of explicit Runge-Kutta methods of or ders p - 1 and p, sets of efficient, continuously differentiable inter polants of several orders up to p are characterized algorithmically in terms of several arbitrary parameters. The approach can be applied in an obvious way to yield interpolants for other types of explicit Rung e-Kutta methods, as well as families of other types of explicit and im plicit methods. Derivative evaluations required for each pair of metho ds are reused, and additional derivative evaluations are selected in a n attempt to minimize the total number of stages required. The analysi s provides a lower bound on the number of stages required, and indicat es, for example, why twelve stages are required to provide interpolant s for eight-stage pairs of orders 5 and 6. In contrast to the eighteen stages used to obtain a known interpolant of order 7 for a pair of me thods of orders 6 and 7, only sixteen stages are required by the propo sed derivation. Details for interpolants of orders p = 8, and p = 9 ar e also given. However, it has not been established that the lower boun d is sharp, so further improvement may be possible.