Abstract
Chapter 2 investigates the adapted Runge–Kutta–Nyström (ARKN) methods proposed by Franco (2002) for the system of oscillatory second-order differential equations y″+ω 2 y=f(y,y′), where ω>0 is the main frequency. Based on the internal stages of the traditional RKN methods, ARKN methods adopt a new form of updates which incorporate the special oscillatory structure of the system. Order conditions for ARKN methods are derived by means of the Nyström tree theory. The symplecticity conditions for ARKN methods are obtained. It is also shown that an ARKN method cannot be symmetric. The effectiveness of a one-stage symplectic ARKN method is illustrated by Duffing equations, the Fermi–Pasta–Ulam problem and the “almost periodic” orbit problem. On the basis of the matrix-variation-of-constants formula established by Wu et al. (2009), multidimensional ARKN methods are developed for the more general oscillatory system y″+My=f(y,y′) with a positive semi-definite (not necessarily symmetric) main frequency matrix M. These methods do not rely on the decomposition of M so that they are applicable to oscillatory systems with a positive semi-definite (but not symmetric) frequency matrix.
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Wu, X., You, X., Wang, B. (2013). ARKN Methods. In: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35338-3_2
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DOI: https://doi.org/10.1007/978-3-642-35338-3_2
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