On efficient frequency-dependent parameters of explicit two-derivative improved Runge-Kutta-Nystr[formula omitted]m method with application to two-body problem

Abstract

Efficient trigonometrically-fitted explicit two-derivative improved Runge–Kutta-Nystro¨m methods with three stage fifth-order, denoted as TFTDIRKN5 method is derived for direct solving special type of second-order ordinary differential equation in the form y′(t)=f(t,y(t)) with oscillatory solution. Order conditions of proposed method that includes previous estimated slopes, k-i are presented through Taylor series expansion and comparison of coefficients with power of h. Second-order initial value problems (IVPs) are integrated exactly with numerical solution in linear composition of set functions eiωt and e-iωtwith ω∈R. Certain coefficients of proposed methods are depend on the principle frequency of the numerical problems for deriving trigonometrically-fitted improved Runge–Kutta-Nystro¨m direct methods with two-derivative term. The proposed method is analysed numerically to prove that it is zero stable, consistent and convergent, which are critical for solving problems effectively. Stability region and error analysis of proposed method are investigated. The numerical tests show that the proposed method performs better in comparison with other existing Runge–Kutta-Nystro¨m methods with similar algebraic order.