The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications

In general, extended Runge-Kutta-Nystrom (ERKN) methods are more effective than traditional Runge-Kutta-Nystrom (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplectic conditions and the symmetric conditions, becomes much more complicated than that for RKN methods. Therefore, it is a bottleneck to construct high-order ERKN methods efficiently. In this paper, we first establish the ERKN group Omega for ERKN methods and the RKN group G for RKN methods, respectively. We then rigorously show that ERKN methods are a natural extension of RKN methods, that is, there exists an epimorphism eta of the ERKN group Omega onto the RKN group G. This epimorphism gives a global insight into the structure of the ERKN group by the analysis of its kernel and the corresponding RKN group G. Meanwhile, we establish a particular mapping phi of G into Omega so that each image element is an ideal representative element of the congruence class in Omega. Furthermore, an elementary theoretical analysis shows that this map phi can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism eta together with its section., we may gain knowledge about the structure of the ERKN group Omega via the RKN group G. In light of the theoretical analysis of this paper, we obtain high-order structure-preserving ERKN methods in an effective way for solving oscillatory Hamiltonian systems. Numerical experiments are carried out and the results are very promising, which strongly support our theoretical analysis presented in this paper. (C) 2016 Elsevier Inc. All rights reserved.