Computers Math. Applic. Vol. 20, No. 1, pp. 15-24, 1990 0097-4943/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1990 Pergamon Press plc A NEW EMBEDDED PAIR OF RUNGE-KUTTA FORMULAS OF ORDERS 5 AND 6 M. CALVO, J. I. MONTIJANO and L. RANDEZ Departamento de Matemfitica Aplicada, Universidad de Zaragoza, 50009 Zaragoza, Espafia (Received 16 June 1989) Abstraet--A new pair of embedded Runge-Kutta (RK) formulas of orders 5 and 6 is presented. It is derived from a family of RK methods depending on eight parameters by using certain measures of accuracy and stability. Numerical tests comparing its efficiency to other formulas of the same order in current use are presented. With an extra function evaluation per step, a C t-continuous interpolant of order 5 can be obtained. 1. INTRODUCTION Many Runge-Kutta (RK) codes for the numerical solution of non-stiff initial value problems in ODEs are based on embedded pairs of RK formulas. Thus RKF45 [1] and its successor DERKF in DEPAC [2] produced by Shampine and Watts use a pair of formulas of orders 4 and 5 due to Fehlberg [3, 4]. The subroutine DVERK [5] produced by Hull et aL is based on a pair of formulas of orders 5 and 6 due to Verner [6]. For computations which require higher accuracy a pair of orders 7 and 8 of Fehlberg [3] has been widely used. Although the above codes showed up very well in extensive numerical computations there are some reasons to think that changing the pairs used would improve the efficiency of the codes. First, Dormand and Prince [7, 8] have presented new pairs of RK formulas which are superior in most respects to those currently in use. On the other hand, there are some applications in which the solution is required at a great many specific points and if a code produces answers at these points by taking them to be mesh points, the code can be very inefficient. Consequently the new RK methods should be able to produce in an inexpensive way approximations to the solution between mesh points. Because of this a number of papers [9-13] proposing interpolants for RK formulas have been recently published. The aim of this paper is to present a new pair of RK formulas of orders 5 and 6 that requires eight function evaluations per step. Some tests showing the efficiency of the new pair relative to other formulas in current use are presented. Furthermore, with an additional function evaluation a fifth order interpolant can be obtained. The paper is organized as follows. In Section 2 a family of RK pairs of orders 5 and 6 depending on eight parameters is constructed. Then in Section 3 these parameters are chosen so that the corresponding method is "optimal" with respect to certain measures of accuracy, efficiency and stability that will be made precise later. In Section 4 it is shown that with an extra function evaluation we can have a Ct-continuous solution of order 5 in the whole integration interval. Finally in Section 5 a summary of the numerical experiments with the non-stiff DETEST [14, 15] problems is presented. 2. THE FAMILY OF EMBEDDED RK FORMULAS Consider the numerical solution of the non-stiff system of ODEs y'(t)=f(t,y(t)), t >~to, y~N, (1) y(to) = Y0, (2) where f(t, y) is assumed to be sufficiently differentiable in a neighborhood of the exact solution (t,y(t)), t ~ [t0, to + T]. An embedded pair of RK formulas is given by two formulas of orders p and q/>p + 1 which share the same function evaluations. In the usual notation, the procedure It