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Explicit Gautschi-type integrators for nonlinear multi-frequency oscillatory second-order initial value problems

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Abstract

The main theme of this paper is explicit Gautschi-type integrators for the nonlinear multi-frequency oscillatory second-order initial value problems of the form \(y^{\prime \prime }= -A(t,y)y+f(t,y),\ y(t_{0}) = y_{0},\ y^{\prime }(t_{0}) = y^{\prime }_{0}\). This work is important and interesting within the broader framework of the subject. In fact, the Gautschi-type methods for oscillatory problems with a constant matrix A have been investigated by many authors. The key question now is that the classical variation-of-constants approach is not applicable to the oscillatory nonlinear problems with a variable coefficient matrix A(t,y). We consider successive approximations or locally equivalent systems for the problems, and derive efficient explicit Gautschi-type integrators. The error analysis is presented for the local approximation accordingly. Accompanying numerical results demonstrate the remarkable efficiency of the new Gautschi-type integrators in comparison with some existing numerical methods in the scientific literature.

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Funding

The research was financially supported in part by the Natural Science Foundation of China under Grants 11501288, 11701271, and 11671200, 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, by the Natural Science Foundation of Jiangsu Province under Grant BK20150934, and by the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grants 16KJB110010 and 14KJB110009.

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Authors and Affiliations

  1. College of Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, People’s Republic of China

    Wei Shi

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, China

    Xinyuan Wu

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

    Xinyuan Wu

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  1. Wei Shi
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  2. Xinyuan Wu
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Correspondence to Wei Shi.

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Shi, W., Wu, X. Explicit Gautschi-type integrators for nonlinear multi-frequency oscillatory second-order initial value problems. Numer Algor 81, 1275–1294 (2019). https://doi.org/10.1007/s11075-018-0635-7

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  • DOI: https://doi.org/10.1007/s11075-018-0635-7

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