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Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t))

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Abstract

In this paper, multi-step hybrid methods for solving special second-order differential equations y ″(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197–220, 8) as special cases. The order conditions of the methods were derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions, we construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison 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 & Xianfen Wang

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

    Jiyong Li & Xianfen Wang

Authors
  1. Jiyong Li
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  2. Xianfen Wang
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Correspondence to Jiyong Li.

Additional information

The research was supported in part by Hebei Natural Science Foundation of China under Grant No: A2014205136, by the Natural Science Foundation of China under Grant No: 11401164, by the Natural Science Foundation of China under Grant No: 11201113 and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant No: 20121303120001.

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Li, J., Wang, X. Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t)). Numer Algor 73, 711–733 (2016). https://doi.org/10.1007/s11075-016-0114-y

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  • DOI: https://doi.org/10.1007/s11075-016-0114-y

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