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A Note on High-Precision Approximation of Asymptotically Decaying Solution and Orthogonal Decomposition

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Abstract

In some physical applications, the decaying rate of asymptotically decaying solution is more important than the solution magnitude itself in understanding the physical system such as the late-time behavior of decaying fields in black hole space-time. In Khanna (J Sci Comput 56(2):366–380, 2013), it was emphasized that high-precision arithmetic and high-order methods are required to capture numerically the correct decaying rate of the late-time radiative tails of black-hole system in order to prevent roundoff errors from inducing a wrong power-law decay rate in the numerical approximation. In this paper, we explain how roundoff errors induce a wrong decay mode in the numerical approximation using simple linear differential equations. Then we describe the orthogonal decomposition method as a possible technique to remove wrong decaying modes induced by roundoff errors in the numerical approximation.

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Acknowledgements

The authors sincerely thank the two anonymous reviewers whose comments helped them to understand the problem better and improve their paper.

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Authors and Affiliations

  1. Department of Mathematics, University at Buffalo, The State University of New York, Buffalo, Buffalo, NY, 14260-2900, USA

    John Nicponski & Jae-Hun Jung

  2. Department of Data Science, Ajou University, Yeongtong-gu, Suwon, 16499, Korea

    Jae-Hun Jung

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  1. John Nicponski
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  2. Jae-Hun Jung
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Correspondence to Jae-Hun Jung.

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Nicponski, J., Jung, JH. A Note on High-Precision Approximation of Asymptotically Decaying Solution and Orthogonal Decomposition. J Sci Comput 76, 189–215 (2018). https://doi.org/10.1007/s10915-017-0619-0

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  • DOI: https://doi.org/10.1007/s10915-017-0619-0

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