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Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems

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Abstract

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y '. In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ?(t) + ? 2 y(t) + µ y '(t)=0 with µ<2?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ?(t) + ? 2 y(t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity.

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Acknowledgments

The research was supported in part by the Natural Science Foundation of China under Grant 11271186, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by the 985 Project at Nanjing University under Grant 9112020301, and by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, by the Natural Science Foundation of Jiangsu Province under Grant BK20150934, by the Natural Science Foundation of China under Grant 11501288, and by project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 16KJB110010.

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Authors and Affiliations

  1. College of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, People’s Republic of China

    Kai Liu

  2. Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China

    Xinyuan Wu

  3. State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, People’s Republic of China

    Xinyuan Wu

  4. College of Mathematical Sciences, Nanjing Tech University, Nanjing, People’s Republic of China

    Wei Shi

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  1. Kai Liu
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  2. Xinyuan Wu
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  3. Wei Shi
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Correspondence to Wei Shi.

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Liu, K., Wu, X. & Shi, W. Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems. Numer Algor 77, 37–56 (2018). https://doi.org/10.1007/s11075-017-0303-3

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  • DOI: https://doi.org/10.1007/s11075-017-0303-3

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