A novel class of Runge-Kutta-Nyström pairs sharing orders 8(6)

We are examining a second-order system of non-stiff Initial Value Problems (IVP), focusing on a scenario where the first derivatives are not present. In the realm of solving IVPs, Runge-Kutta-Nyström (RKN) pairs have proven to be highly effective. In order to achieve a pair with eighth and sixth order accuracy, we need to find a solution to a well-defined set of equations regarding the coefficients. Traditionally, pairs are constructed to go through eight stages per step. However, we propose a novel approach with nine stages per step, which enables the creation of pairs with orders $8$ and $6$ that have notably smaller truncation errors. Our paper introduces a new pair that, as expected, outperforms existing pairs of the same orders in a range of important problems.