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Error analysis of explicit TSERKN methods for highly oscillatory systems

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Abstract

In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486–2507] for multi-frequency and multidimensional oscillatory systems y″(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197–220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, China

    Jiyong Li

  2. Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024, China

    Jiyong Li

  3. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Jiyong Li & Xinyuan Wu

  4. State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210093, China

    Jiyong Li

Authors
  1. Jiyong Li
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  2. Xinyuan Wu
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Correspondence to Xinyuan Wu.

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Dedicated to John’s 80th Birthday

The research was supported by the Natural Science Foundation of China under Grant 11271186, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant20100091110033, by the 985 Project at Nanjing University under Grant 9112020301, by the Fundamental Research Funds for the Central Universities, and by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Li, J., Wu, X. Error analysis of explicit TSERKN methods for highly oscillatory systems. Numer Algor 65, 465–483 (2014). https://doi.org/10.1007/s11075-013-9761-4

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  • DOI: https://doi.org/10.1007/s11075-013-9761-4

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