New optimized symmetric and symplectic trigonometrically fitted RKN methods for second-order oscillatory differential equations

Symmetric, symplectic and trigonometrically fitted Runge–Kutta–Nyström (SSTFRKN) methods for second-order differential equations with oscillatory solutions are investigated. Symmetry, symplecticity and trigonometric fitting conditions for modified Runge–Kutta–Nyström (RKN) methods are presented. Order conditions for modified RKN methods are derived via the special Nyström tree theory. Two explicit SSTFRKN methods with variable nodes are derived. The two new methods are zero-dissipative due to symplecticity. Their dispersion orders are analysed and their periodicity regions are obtained. The results of numerical experiments show the robustness and competence of the new SSTFRKN methods compared with some highly efficient codes in the recent literature.