For the integration of multi-frequency second-order oscillatory systems of differential equations, multidimensional extended Runge–Kutta–Nyström (MERKN) methods with FSAL property are investigated. Order conditions are presented. Phase properties are analysed and nonlinear stability of MERKN methods is proved. New practical MERKN methods with optimized phase and dissipation properties and embedded pairs of multidimensional MERKN methods are constructed. Numerical experiments are reported to show the high efficiency and robustness of our new methods compared with several well-known methods in the scientific literature.

中文翻译：

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多频振荡系统具有 FSAL 属性的多维 ERKN 方法的优化对

对于微分方程的多频二阶振荡系统的积分，研究了具有 FSAL 特性的多维扩展 Runge-Kutta-Nyström (MERKN) 方法。提供订单条件。分析了相性质并证明了MERKN方法的非线性稳定性。构建了具有优化相位和耗散特性的新实用 MERKN 方法以及嵌入的多维 MERKN 方法对。据报道，与科学文献中的几种众所周知的方法相比，数值实验表明我们的新方法具有高效率和稳健性。