Construction of an optimized explicit Runge–Kutta–Nyström method for the numerical solution of oscillatory initial value problems

An explicit optimized Runge–Kutta–Nyström method with four stages and fifth algebraic order is developed. The produced method has variable coefficients with zero phase-lag, zero amplification factor and zero first derivative of the amplification factor. We provide an analysis of the local truncation error of the new method. We also measure the efficiency of the new method in comparison to other numerical methods through the integration of the two-body problem with various eccentricities and three other periodical/oscillatory initial value problems.