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

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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2017, Volume 57, Number 1, Pages 9–28
DOI: https://doi.org/10.7868/S0044466917010057 (Mi zvmmf10503)

This article is cited in 21 scientific papers (total in 21 papers)

Cubic spline interpolation of functions with high gradients in boundary layers

I. A. Blatov^a, A. I. Zadorin^b, E. V. Kitaeva^c

^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

Full-text PDF (338 kB) Citations (21)

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

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

Received: 03.02.2016
Revised: 31.03.2016

English version:
Computational Mathematics and Mathematical Physics, 2017, Volume 57, Issue 1, Pages 7–25
DOI: https://doi.org/10.1134/S0965542517010043

Bibliographic databases:

Document Type: Article

UDC: 519.652.3

Language: Russian

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

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Linking options:

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This publication is cited in the following 21 articles:

I.A. Blatov, A I Zadorin, “Analysis of approaches to spline interpolation of functions with large gradients in the boundary layer”, J. Phys.: Conf. Ser., 2182:1 (2022), 012016
A. I. Zadorin, “Two-Dimensional Interpolation of Functions by Cubic Splines in the Presence of Boundary Layers”, J Math Sci, 267:4 (2022), 511
Igor Blatov, Elena Kitaeva, Nikita Zadorin, Lecture Notes in Computational Science and Engineering, 141, Mesh Methods for Boundary-Value Problems and Applications, 2022, 39
Gang Lu, Yide Liu, Xue Yao, Jiachen Yang, Cheng Jia, Arpit Bhardwaj, “Exploration of Laser Marking Path and Algorithm Based on Intelligent Computing and Internet of Things”, Computational Intelligence and Neuroscience, 2022 (2022), 1
W. Pan, X. Luo, M. Zhu, J. Ye, L. Gong, H. Qu, “A health indicator extraction and optimization for capacity estimation of Li-ion battery using incremental capacity curves”, J. Energy Storage, 42 (2021), 103072
J.-p. Wang, Sh.-h. Wang, Ya.-q. Wang, H. Hu, J.-w. Yu, X. Zhao, J.-l. Liu, X. Chen, Yu. Li, “A data process of human knee joint kinematics obtained by motion-capture measurement”, BMC Med. Inform. Decis. Mak., 21:1 (2021), 121
I A Blatov, N V Dobrobog, E V Kitaeva, “The cubic interpolation spline for functions with boundary layer on a Bakhvalov mesh”, J. Phys.: Conf. Ser., 1715:1 (2021), 012001
I A Blatov, A I Zadorin, “Application a cubic spline to calculate derivatives in the presence of a boundary layer”, J. Phys.: Conf. Ser., 1791:1 (2021), 012069
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Application of cubic splines on Bakhvalov meshes in the case of a boundary layer”, Comput. Math. Math. Phys., 61:12 (2021), 1911–1930
I. A. Blatov, N. V. Dobrobog, E. V. Kitaeva, “The development of V. V. Strygin's ideas in numerical mathematics in the works of Samara mathematicians”, Applied Mathematics, Computational Science and Mechanics: Current Problems, Journal of Physics Conference Series, 1479, IOP Publishing Ltd, 2020, 012102
B. Su, R. Rao, Zh. Li, L. Song, J. Yue, “Detecting permafrost in plateau and mountainous areas by airborne transient electromagnetic sensing”, Electronics, 9:8 (2020), 1229
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Generalized spline interpolation of functions with large gradients in boundary layers”, Comput. Math. Math. Phys., 60:3 (2020), 411–426
X. Liu, X. Luan, Ya. Yin, F. Liu, “Feature-based data alignment of multi-stage batch processes and its application to optimization”, IFAC-PapersOnLine, 52:1 (2019), 778–783
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “An application of the cubic spline on Shishkin mesh for the approximation of a function and its derivatives in the presence of a boundary layer”, XII International Scientific and Technical Conference Applied Mechanics and Systems Dynamics, Journal of Physics Conference Series, 1210, IOP Publishing Ltd, 2019, 012017
I. A. Blatov, A. I. Zadorin, “Approaches to the calculation of derivatives of functions with large gradients in the boundary layer under the values at the grid nodes”, 12Th International Conference - Mesh Methods For Boundary: Value Problems and Applications, Journal of Physics Conference Series, 1158, IOP Publishing Ltd, 2019, 022029
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Approximation of a function and its derivatives on the basis of cubic spline interpolation in the presence of a boundary layer”, Comput. Math. Math. Phys., 59:3 (2019), 343–354
Alexander Zadorin, Igor' Blatov, Lecture Notes in Computer Science, 11386, Finite Difference Methods. Theory and Applications, 2019, 654
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
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
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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