A Novel Explicit Two-Derivative Runge–Kutta–Nyström Method with Energy Conservation for the Integration of Second-Order Periodic ODEs

Abstract

A novel trigonometrically-fitted explicit two-derivative Runge-Kutta-Nyström (TFETDRKN(5)) method with three-stage and fifth-order for solving a class of special second-order (system) ODEs in the form of $u'' = f\left( {t,u} \right)$ with periodic solutions is proposed. Order conditions of the new explicit two-derivative Runge-Kutta-Nyström (ETDRKN(5)) method are derived using Taylor expansion and comparison of step size, $h$ over Taylor method and general formula of the ETDRKN(5) method. Trigonometrically-fitting technique is implemented into the ETDRKN(5) method to form the TFETDRKN(5) method. Stability analysis of the new proposed method is thoroughly investigated and discussed. Algebraic order of the ETDRKN(5) method is investigated. Numerical experiments for the TFETDRKN(5) method are conducted versus error accuracy, number of function evaluations and computational time. Numerical tables and graphs demonstrate that the TFETDRKN(5) method has higher effectiveness and accuracy compared to selected existing methods. Further study for one typical real-word experiment is conducted. Besides, Hamiltonian energy, Lagrangian energy and momentum conservation of proposed method are investigated to outlook the energy conservation property. The related energy exchange of the above three energies during the tested real-world experiment is illustrated.