Extended Runge‐Kutta‐Nyström Methods Based on Hermite Interpolation for Second‐Order Oscillatory Systems

Extended Runge‐Kutta‐Nyström methods based on Hermite interpolation (ERKNH) for the numerical solution of general second‐order oscillatory systems of the form are proposed and investigated. It is proved that the ‐stage ERKNH methods achieve the optimal convergence order . The nonlinear stability of the newly constructed methods is analyzed, and their symmetry property is discussed. Four representative new ERKNH methods are derived and numerical experiments are carried out to illustrate the high efficiency of the proposed methods.