Continuous-Stage ERKN Integrators for Second-Order ODEs with Highly Oscillatory Solutions

In this chapter, continuous-stage extended Runge–Kutta–Nystrom (CSERKN) integrators for solving highly oscillatory systems of second-order ODEs are derived and analysed. These integrators are incorporated into the special structure of highly oscillatory systems so that their internal stages and updates can integrate the associated highly oscillatory homogeneous systems exactly. When the underlying highly oscillatory systems are Hamiltonian systems, sufficient conditions for energy preservation are shown for CSERKN methods. The symmetry and stability of CSERKN integrators are also analysed in detail. Preliminary numerical results highlight the effectiveness of CSERKN methods.