Diagonal implicit symplectic extended RKN methods for solving oscillatory Hamiltonian systems

This paper studies diagonal implicit symplectic extended Runge–Kutta–Nystrom (ERKN) methods for solving the oscillatory Hamiltonian system \(H(q,p)=\dfrac{1}{2}p^\mathrm{T}p+\dfrac{1}{2}q^\mathrm{T}Mq+U(q)\). Based on symplecticity conditions and order conditions, we construct some diagonal implicit symplectic ERKN methods. The stability of the obtained methods is discussed. Three numerical experiments are carried out to show the performance of the methods. It follows from the numerical results that the new diagonal implicit symplectic methods are more effective than RKN methods when applied to the oscillatory Hamiltonian system.