Abstract

Runge – Kutta – Nystrom pairs of orders 5 and 4 are well suited for the solution of Initial value problem of second order. During the last decades were pushed aside by the pairs of orders 6 and 4 share the same number cost. Namely, 5 stages per step. Here we propose an alternative case of such pairs using only four stages per step. This was achieved by solving the corresponding equations of condition by the technique of Differential Evolution. Finally some numerical tests justify our effort.

Highlights

  • œ f i f (x n c h,y in n ci h n ya n hn2 a f ), ij j j 1 where s the number of stages per step and A ‰ \sqs, bT,c ‰ \s the coefficients of the method

  • Along with the above formula we may deduce another approximation of order q p at no cost

  • Bettis [7] published among the first such a pair at a cost of 5 stages per step

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Summary

Introduction

Explicit Runge – Kutta – Nyström pairs [1,2,3,4] are amongst the most popular methods for integrating Initial Value Problems of second order with the special form, y aa f (x, y), y(x ) y , y a(x ) y a. œ f i f (x n c h ,y in n ci h n ya n hn a f ), ij j j 1 where s the number of stages per step and A ‰ \sqs , bT ,c ‰ \s the coefficients of the method. The above formula is equivalent to Taylor series expansion of order p. Otherwise the advance from x to n x x h is aborted and the new smaller step n 1 n n h h n n 1 is used for advancing from x n to x n 1 x n. For details see [5]

The new method
Numerical Tests
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