Runge–Kutta–Nyström methods of eighth order for addressing Linear Inhomogeneous problems

Second order Linear Inhomogeneous Initial Value Problems with constant coefficients are considered here. Runge–Kutta–Nyström (RKN) methods are amongst the most effective for addressing these problems. For attaining a certain algebraic order of accuracy, we need to solve a reduced set of order conditions (in comparison with conventional RKN) with respect to the coefficients. Here, using a seven stages method we manage to solve these equations up to eighth order. Differential Evolution technique was used for achieving this. Thus, our main contribution is that we saved one stage since conventional eight order RKN methods use eight stages for advancing a step. This new method is combined at no cost with a fifth and third order companion formulas. As consequence, a triplet of orders 8(5)3 is given for estimating local errors and adjust effectively the step lengths. The new triplet of methods is applied to various relevant problems and is found to outperform standard RKN pairs from the literature.