Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems

The ERKN methods proposed by H. Yang et al. [Comput. Phys. Comm. 180 (2009) 1777] are an important improvement of J.M. Francoʼs ARKN methods for perturbed oscillators [J.M. Franco, Comput. Phys. Comm. 147 (2002) 770]. This paper focuses on the symmetry and symplecticity conditions for ERKN methods solving oscillatory Hamiltonian systems. Two examples of symmetric and symplectic ERKN (SSERKN) methods of orders two and four respectively are constructed. The phase and stability properties of the new methods are analyzed. The results of numerical experiments show the robustness and competence of the SSERKN methods compared with some well-known methods in the literature.