Interpolation Formulas for Functions with Large Gradients in the Boundary Layer and their Application

Задорин, А. И.

doi:10.18255/1818-1015-2016-3-377-384

Cited by 4 publications

References 5 publications

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Analysis of the numerical differentiation formulas of functions with large gradients

Tikhovskaya

2017

AIP Conference Proceedings

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A two-grid method with Richardson extrapolation for a semilinear convection-diffusion problem AIP Conference Proceedings 1684, 090007 (2015) Abstract. The solution of a singularly perturbed problem corresponds to a function with large gradients. Therefore the question of interpolation and numerical differentiation of such functions is relevant. The interpolation based on Lagrange polynomials on uniform mesh is widely applied. However, it is known that the use of such interpolation for the function with large gradients leads to estimates that are not uniform with respect to the perturbation parameter and therefore leads to errors of order O(1). To obtain the estimates that are uniform with respect to the perturbation parameter, we can use the polynomial interpolation on a fitted mesh like the piecewise-uniform Shishkin mesh or we can construct on uniform mesh the interpolation formula that is exact on the boundary layer components. In this paper the numerical differentiation formulas for functions with large gradients based on the interpolation formulas on the uniform mesh, which were proposed by A.I. Zadorin, are investigated. The formulas for the first and the second derivatives of the function with two or three interpolation nodes are considered. Error estimates that are uniform with respect to the perturbation parameter are obtained in the particular cases. The numerical results validating the theoretical estimates are discussed.

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Analysis of the numerical differentiation formulas of functions with large gradients

Tikhovskaya

2017

AIP Conference Proceedings

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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

Blatov

Задорин

Kitaeva

2019

J. Phys.: Conf. Ser.

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Adaptive formulas of numerical differentiation of functions with large gradients

Ильин

Задорин

2019

J. Phys.: Conf. Ser.

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There are constructed formulas for the numerical differentiation of functions of one variable with large gradients. Formulas are based on the fitting to the singular component responsible for the large gradients of the function. The singular component is considered as a function of the general form. The error estimates of the difference formulas for calculating first and second derivatives are obtained. The results of the calculations are given.

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Interpolation Formulas for Functions with Large Gradients in the Boundary Layer and their Application

Cited by 4 publications

References 5 publications

Analysis of the numerical differentiation formulas of functions with large gradients

Analysis of the numerical differentiation formulas of functions with large gradients

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

Adaptive formulas of numerical differentiation of functions with large gradients

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