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Two-Step Multidimensional ERKN Methods

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Structure-Preserving Algorithms for Oscillatory Differential Equations

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Abstract

In Chap. 5, multidimensional two-step extended Runge–Kutta–Nyström-type (TSERKN) methods are developed for solving the oscillatory second-order system y″+My=f(x,y), where M∈ℝd×d is a symmetric positive semi-definite matrix that implicitly contains the frequencies of the problem. The new methods inherit the framework of two-step hybrid methods and are adapted to the special features of the true flows in both the internal stages and the updates. Based on the SEN-tree theory in Chap. 3, order conditions for the TSERKN methods are derived via the B-series defined on the set SENT of trees and the B f-series defined on the subset SENT f of SENT. Three explicit TSERKN methods are constructed and their stability and phase properties are analyzed. Numerical experiments show the applicability and efficiency of the new methods in comparison with the well-known high quality methods proposed in the literature.

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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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  3. Bin Wang
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© 2013 Science Press Beijing and Springer-Verlag Berlin Heidelberg

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Wu, X., You, X., Wang, B. (2013). Two-Step Multidimensional ERKN Methods. In: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35338-3_5

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  • DOI: https://doi.org/10.1007/978-3-642-35338-3_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35337-6

  • Online ISBN: 978-3-642-35338-3

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