Relaxation RKN-type integrators that preserve two invariants for second-order (oscillatory) systems

Recently, the relaxation technique has been widely used to impose conservation of invariants while retaining the full accuracy of the original method. So far, only a single invariant of a system has been considered. In this work, by a mild generalization of the relaxation technique, the Runge–Kutta–Nyström (RKN) integrators are modified to preserve two invariants for second-order system of Ordinary Differential Equations (ODEs). The proposed integrators can be explicit and of arbitrarily high order. The accuracy of the relaxation RKN integrators and the existence of valid relaxation parameters have been proved. The construction of the new integrators is under the framework of adapted RKN (ARKN) integrators which are specially designed for numerical solving second-order oscillatory systems. Therefore, the proposed integrators could be oscillation-preserving in the sense that they exactly integrate homogeneous oscillatory system $q^{\prime \prime }+Kq=0$. Some numerical experiments are conducted to show the advantage and efficiency of the proposed integrators in comparison with the standard (A)RKN integrators.