Block embedded explicit runge-kutta methods

Abstract

Embedded explicit Runge—Kutta formulae are amongst the most popular methods currently in use for the approximate numerical integration of nonstiff systems of ordinary differential equations. In particular the fourth-order methods RKF45 of Shampine and Watts and the Runge—Kutta—Merson code of the NAG Library have been particularly popular for some time. In this paper we consider the derivation of a block embedded explicit Runge—Kutta (BERK) formula of order 4. BERK formulae have all the characteristics of standard explicit Runge—Kutta formulae except that they are no longer single step in nature in the sense that a pth order BERK formula produces pth order approximations to the solution at several step points instead of at one point only. The results of some fairly extensive numerical testing are presented and these indicate that the new formula is very competitive, both in terms of efficiency and reliability, with the codes R—K Merson of NAG and RKF45. Also considered in detail are the problems of computing solutions at “off-step” points and of efficiently computing low accuracy solutions. In particular we show that BERK formulae allow intermediate solutions at “off-step” points to be computed with relatively little additional computational effort. This makes BERK formulae particularly attractive for problems where output is requested at many off-step points since it is for this class of problems that conventional Runge—Kutta formulae can become very inefficient. Also considered is the question of computing low accuracy solutions. It is shown that, as is to be expected, such problems are solved more efficiently using low order BERK formulae. In particular, BERK formulae of orders 1 and 3 are derived and their performance is compared with that of the fourth-order BERK formula on a large set of test problems.