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An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems

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Abstract

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used.

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Authors and Affiliations

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

    Xianyang Zeng, Hongli Yang & Xinyuan Wu

  2. Industrial Center, Nanjing Institute of Technology, Nanjing, People’s Republic of China

    Hongli Yang

  3. Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, People’s Republic of China

    Xianyang Zeng

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

    Xinyuan Wu

Authors
  1. Xianyang Zeng
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  2. Hongli Yang
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  3. Xinyuan Wu
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Correspondence to Xinyuan Wu.

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Zeng, X., Yang, H. & Wu, X. An improved tri-coloured rooted-tree theory and order conditions for ERKN methods for general multi-frequency oscillatory systems. Numer Algor 75, 909–935 (2017). https://doi.org/10.1007/s11075-016-0225-5

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  • DOI: https://doi.org/10.1007/s11075-016-0225-5

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