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Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 1, Pages 9–28 (Mi zvmmf10503)  

This article is cited in 5 scientific papers (total in 5 papers)

Cubic spline interpolation of functions with high gradients in boundary layers

I. A. Blatova, A. I. Zadorinb, E. V. Kitaevac

a Volga State University of Telecommunications and Informatics, Samara, Russia
b Sobolev Institute of Mathematics (Omsk Branch), Siberian Branch, Russian Academy of Sciences, Omsk, Russia
c Samara State University, Samara, Russia

Abstract: The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes $N$ is fixed. A modified cubic interpolation spline is proposed, for which $O((\ln N/N)^4)$ error estimates that are uniform with respect to the small parameter are obtained.

Key words: singular perturbation, boundary layer, Shishkin mesh, cubic spline, modification, error estimate.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-06584_а
16-01-00727_а


DOI: https://doi.org/10.7868/S0044466917010057

Full text: PDF file (338 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:1, 7–25

Bibliographic databases:

UDC: 519.652.3
Received: 03.02.2016
Revised: 31.03.2016

Citation: I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers”, Zh. Vychisl. Mat. Mat. Fiz., 57:1 (2017), 9–28; Comput. Math. Math. Phys., 57:1 (2017), 7–25

Citation in format AMSBIB
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\paper Cubic spline interpolation of functions with high gradients in boundary layers
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\pages 9--28
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\pages 7--25
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “On the parameter-uniform convergence of exponential spline interpolation in the presence of a boundary layer”, Comput. Math. Math. Phys., 58:3 (2018), 348–363  mathnet  crossref  crossref  isi  elib
    2. B. Wang, X. Gu, Sh. Yan, “STCS: a practical solar radiation based temperature correction scheme in meteorological WSN”, Int. J. Sens. Netw., 28:1 (2018), 22–33  crossref  isi
    3. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “An application of the exponential spline for the approximation of a function and its derivatives in the presence of a boundary layer”, Mechanical Science and Technology Update (MSTU-2018), Journal of Physics Conference Series, 1050, IOP Publishing Ltd, 2018, 012012  crossref  mathscinet  isi  scopus
    4. V A. Irina, N. Victor Ignat'ev, “Modeling intermolecular interaction in multicomponent environments in energy power systems”, 2018 International Scientific Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon), IEEE, 2018  crossref  isi
    5. Liu X., Luan X., Yin Ya., Liu F., “Feature-Based Data Alignment of Multi-Stage Batch Processes and Its Application to Optimization”, IFAC PAPERSONLINE, 52:1 (2019), 778–783  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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