PII : S 0898 - 1221 ( 96 ) 00141 . 1 An Efficient Runge - Kutta ( 4 , 5 ) Pair
نویسنده
چکیده
-A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step. Keywords--Ordinary differential equations, Runge-Kutta, RKSUITE, Continuous extension, Local error. 1. I N T R O D U C T I O N Fourth-order explicit Runge-Kutta formulas have always been popular for the solution of the initial value problem for a first-order system of ordinary differential equations (ODEs) y' = f ( x , y ) , a < x < b, y(a) given. The error made in a step, the local error, is estimated by taking each step with a fifth-order formula and estimating the error in the fourth-order formula by comparison. A natural measure of the cost of a Runge-Kutta formula is the number of stages involved--the number of times f(x, 9) is evaluated. By embedding the evaluation of one formula in the other, it is possible to make evaluation of the pair very much cheaper than separate evaluation of the individual formulas. At least six stages are needed for a fifth-order formula, and it is possible to derive pairs that require only six stages. The landmark paper of Hull et al. [i] considers how to assess the effectiveness of methods for the numerical solution of ODEs. There, the six stage F(4,5) pair due to Fehlberg [2] proved to be very effective. Provided that the stability of the fifth-order formula is acceptable, advancing the integration with the higher-order result, called local extrapolation [3], results in a more accurate integration at no additional cost. The comparison [4] shows the considerable advantages of implementing F(4,5) in this way. For quite some time the F(4,5) pair in local extrapolation mode The authors are grateful for the advice of I. Gladwell that has considerably improved this paper.
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