Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems

This paper studies explicit multi-frequency symmetric extended Runge–Kutta–Nyström (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations q̋(t)+Mq(t)=f(q(t)). We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations.