An embedded explicit Runge–Kutta–Nyström method for solving oscillatory problems

Abstract

A new pair of embedded explicit Runge–Kutta–Nystrom (RKN) methods is developed to integrate second-order differential equations of the form q'' =f(t,q) where the solution is oscillatory. The embedded formula has dispersion order eight and dissipation order seven for the fifth-order formula. The cost for this pair is four function evaluations at each step of integration. Numerical comparisons with several codes in the scientific literature such as RKN5(4)D, RKN5(4)B, RKN4(3)G and DOPRI5 show the efficiency of the new method developed.