A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only symmetric but also symplectic integrators. In analysis, we neither assume symplecticity methods, nor symmetry methods. It turns out that these both types have near total and over long term. To prove result explicit integrators, relationship between ERKN trigonometric is established. For implicit above approach does work anymore use technology modulated Fourier expansion. By taking some adaptations this derive expansion show conservation.