Abstract:
The problem of numerical differentiation of functions with large gradients in the boundary layer is investigated. The problem is that in the case of functions with large gradients and a uniform grid, the relative error of the classical difference formulas for derivatives can be significant. It is proposed to use the Shishkin mesh to obtain a relative error of the formulas independent of a small parameter. Error estimates that depend on the number of nodes of the difference formulas for a derivative of a given order are obtained. It is proved that the error estimate is uniform in terms of a small parameter. In the case of the uniform grid, the region of the boundary layer is allocated, outside of which the numerical differentiation formulas have an error that is uniform in terms of a small parameter. The results of the numerical experiments are presented.
Citation:
A. I. Zadorin, “The analysis of numerical differentiation formulas on the Shishkin mesh with of a boundary layer”, Sib. Zh. Vychisl. Mat., 21:3 (2018), 243–254; Num. Anal. Appl., 11:3 (2018), 193–203
\Bibitem{Zad18}
\by A.~I.~Zadorin
\paper The analysis of numerical differentiation formulas on the Shishkin mesh with of a~boundary layer
\jour Sib. Zh. Vychisl. Mat.
\yr 2018
\vol 21
\issue 3
\pages 243--254
\mathnet{http://mi.mathnet.ru/sjvm681}
\crossref{https://doi.org/10.15372/SJNM20180301}
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\transl
\jour Num. Anal. Appl.
\yr 2018
\vol 11
\issue 3
\pages 193--203
\crossref{https://doi.org/10.1134/S1995423918030011}
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Linking options:
https://www.mathnet.ru/eng/sjvm681
https://www.mathnet.ru/eng/sjvm/v21/i3/p243
This publication is cited in the following 9 articles:
A. I. Zadorin, “Formulas for Numerical Differentiation on a Uniform Mesh in the Presence of a Boundary Layer”, Comput. Math. and Math. Phys., 64:6 (2024), 1167
A. I. Zadorin, “Analiz formul chislennogo differentsirovaniya na ravnomernoi setke pri nalichii pogranichnogo sloya”, Tr. IMM UrO RAN, 30, no. 4, 2024, 106–116
A. I. Zadorin, “Analysis of Numerical Differentiation Formulas on a Uniform Grid in the Presence of a Boundary Layer”, Proc. Steklov Inst. Math., 327:S1 (2024), S275
A. I. Zadorin, “Formuly chislennogo differentsirovaniya funktsii s bolshimi gradientami”, Sib. zhurn. vychisl. matem., 26:1 (2023), 17–26
A. I. Zadorin, “Analysis of numerical differential formulas on a Bakhvalov mesh in the presence of a boundary layer”, Comput. Math. Math. Phys., 63:2 (2023), 175–183
N. A. Zadorin, “Analiz formul chislennogo differentsirovaniya funktsii s bolshimi gradientami na setke Bakhvalova”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 163, no. 3, Izd-vo Kazanskogo un-ta, Kazan, 2021, 261–275
A. I. Zadorin, N. A. Zadorin, “Non-polynomial interpolation of functions with large gradients and its application”, Comput. Math. Math. Phys., 61:2 (2021), 167–176
N. A. Zadorin, “Numerical differentiation on the bakhvalov mesh in the presence of an exponential boundary layer”, Iv International Scientific and Technical Conference Mechanical Science and Technology Update (Mstu-2020), Journal of Physics Conference Series, 1546, IOP Publishing Ltd, 2020, 012108
I. A. Blatov, N. A. Zadorin, “Interpolyatsiya na setke Bakhvalova pri nalichii eksponentsialnogo pogranichnogo sloya”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 161, no. 4, Izd-vo Kazanskogo un-ta, Kazan, 2019, 497–508
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