The Construction of Arbitrary Order ERKN Integrators via Group Theory

This chapter presents the construction of arbitrary order extended Runge–Kutta–Nyström (ERKN) integrators. In general, ERKN methods are more effective than traditional Runge–Kutta–Nyström (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplecticity 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. This chapter first establishes the ERKN group $$\varOmega $$ for ERKN methods and the RKN group G for RKN methods, respectively, and then shows that ERKN methods are a natural extension of RKN methods. That is, there exists an epimorphism $$\eta $$ of the ERKN group $$\varOmega $$ 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. We also establish a particular mapping $$\varphi $$ of G into $$\varOmega $$ that each image element is an ideal representative element of the congruence class in $$\varOmega $$ . Furthermore, an elementary theoretical analysis shows that this mapping $$\varphi $$ can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism $$\eta $$ together with its section $$\varphi $$ , we may gain knowledge about the structure of the ERKN group $$\varOmega $$ through the RKN group G.