Development of high-order adaptive multi-step Runge–Kutta–Nyström method for solving special second-order ODEs

Runge–Kutta–Nyström (RKN) methods are extensively used to obtain approximate solutions of ordinary differential equations (ODEs). Specifically, they are widely used to directly solve second-order ODEs of the special form. Although the derivation of new higher-order methods with fewer numbers of function evaluations is of great importance in increasing the precision and effectiveness of the methods, however, this is rarely done due to the difficulty or complexity of some derivations. This study focuses on constructing a 7(5) pair of embedded multi-step Runge–Kutta–Nyström (EMSN) method with lower stages for the numerical solutions of special second-order ODEs. An adaptive step size formulation using an embedded procedure is considered, and the numerical findings reveal that the new embedded pair outperforms existing Runge–Kutta (RK) pairs in terms of the minimum number of functions evaluations.