An optimized explicit Runge–Kutta–Nyström method for the numerical solution of orbital and related periodical initial value problems

In this work a procedure for the construction of an explicit optimized Runge–Kutta–Nyström method with four stages and fifth algebraic order is provided. The variable coefficients of the preserved method result after nullifying the phase-lag, the dissipative error and the first derivative of the phase-lag. We can see the efficiency of the new method through its local truncation error. Furthermore, we compare the new methodʼs efficiency to other numerical methods. This is shown through the integration of the two-body problem with various eccentricities and of four other initial value problems.