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Symplectic and Symmetric Multidimensional ERKN Methods

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Structure-Preserving Algorithms for Oscillatory Differential Equations

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Abstract

Symplectic and symmetric Runge–Kutta–Nyström-type methods have been shown to have excellent behavior in the long-term integration of Hamiltonian systems. Chapter 4 focuses on the investigation of symplectic and symmetric multi-frequency and multidimensional extended Runge–Kutta–Nyström (SSMERKN) integrators. The symplecticity and symmetry conditions for multidimensional ERKN methods are obtained. When the principal frequency matrix vanishes, they reduce to those for the traditional RKN methods with constant coefficients. Some practical SSMERKN integrators are derived. The stability and phase properties of SSMERKN integrators are analyzed. A technique is developed for transforming a non-autonomous Hamiltonian system into an equivalent autonomous Hamiltonian system in an extended phase space. Symplectic multidimensional ERKN methods applied to the equivalent system are shown to preserve the extended energy very well. Numerical experiments are carried out on three nonlinear wave equations and the Fermi–Pasta–Ulam problem.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-642-35338-3_10

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Authors and Affiliations

  1. Nanjing University, Nanjing, China

    Xinyuan Wu & Bin Wang

  2. Nanjing Agricultural University, Nanjing, China

    Xiong You

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  1. Xinyuan Wu
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  2. Xiong You
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Wu, X., You, X., Wang, B. (2013). Symplectic and Symmetric Multidimensional ERKN Methods. In: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35338-3_4

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  • DOI: https://doi.org/10.1007/978-3-642-35338-3_4

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