Adaptive multi-step Runge–Kutta–Nyström methods for general second-order ordinary differential equations

In this study, a variable step size formulation of multi-step general Runge–Kutta–Nyström (MSGN) methods to directly integrate general second-order initial value problems (IVPs) is considered. This formula is carried out using an embedded explicit pair where: the higher-order formula is an accurate and the lower-order formula uses to estimate the local error. Numerical results show that the new methods perform much better in terms of functions evaluations, implementation cost and execution time compared to other existing high quality embedded Runge–Kutta (ERK) methods in the literature.