The analysis of numerical differentiation formulas on the Shishkin mesh with of a boundary layer
A.I. Zadorin
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090
Keywords: функция одной переменной, пограничный слой, формула численного дифференцирования, сетка Шишкина, оценка погрешности, one-variable function, boundary layer, numerical differentiation formula, Shishkin mesh, error estimate
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.
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