Two-step extended RKN methods for oscillatory systems

In this paper, two-step extended Runge–Kutta–Nyström-type methods for the numerical integration of perturbed oscillators are presented and studied. The new methods inherit the framework of two-step hybrid methods and are adapted to the special feature of the true flows in both the internal stages and the updates. Based on the EN-trees theory [H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777–1794], order conditions for the new methods are derived via the BBTBBT-series defined on the set BT   of branches and the BBWTBBWT-series defined on the subset BWT of BT. The stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of our new methods in comparison with the well-known high quality methods proposed in the scientific literature.

► Two-step ERKN methods (TSERKN) for oscillatory systems are proposed.
► Order conditions for two-step ERKN methods are presented based on the BBTBBT-series.
► The explicit TSERKN methods are constructed via the order condition derived in this paper.
► The efficiency of the new methods is shown in comparison with the high quality codes proposed in the scientific literature.