A FAMILY OF 5TH-ORDER RUNGE-KUTTA PAIRS

The construction of a Runge-Kutta pair of order 5(4) with the minimal number of stages requires the solution of a nonlinear system of 25 ord er conditions in 27 unknowns. We define a new family of pairs which in cludes pairs using 6 function evaluations per integration step as well as pairs which additionally use the first function evaluation from th e next step. This is achieved by making use of Kutta's simplifying ass umption on the original system of the order conditions, i.e., that all the internal nodes of a method contributing to the estimation of the endpoint solution provide, at these nodes, cost-free second-order appr oximations to the true solution of any differential equation. In both cases the solution of the resulting system of nonlinear equations is c ompletely classified and described in terms of five free parameters. O ptimal Runge-Kutta pairs with respect to minimized truncation error co efficients; maximal phase-lag order and various stability characterist ics are presented. These pairs were selected under the assumption that they are used in Local Extrapolation Mode (the propagated solution of a problem is the one provided by the fifth-order formula of the pair) . Numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency w hen using a specific pair of the new family with minimized truncation error coefficients, instead of some other existing pairs.