This article is cited in 8 papers

Approximation of a function and its derivatives on the basis of cubic spline interpolation in the presence of a boundary layer

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

a</sup> Povolzhskiy State University of Telecommunications and Informatics, Samara, 443010 Russia
^b Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
^c Samara University, Samara, 443086 Russia

Abstract: The problem of approximate calculation of the derivatives of functions with large gradients in the region of an exponential boundary layer is considered. It is known that the application of classical formulas of numerical differentiation to functions with large gradients in a boundary layer leads to significant errors. It is proposed to interpolate such functions by cubic splines on a Shishkin grid condensed in the boundary layer. The derivatives of a function defined on the grid nodes are found by differentiating the cubic spline. Using this approach, estimates of the relative error in the boundary layer and the absolute error outside of the boundary layer are obtained. These estimates are uniform in a small parameter. The results of computational experiments are discussed.

Key words: function of one variable, exponential boundary layer, Shishkin grid, cubic spline, approximation of derivatives, error estimate.

UDC: 519.651

Received: 04.07.2018

DOI: 10.1134/S0044466919030049

 English version: Computational Mathematics and Mathematical Physics, 2019, 59:3, 343–354

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