Formulas for numerical differentiation of functions with large gradients
A.I. Zadorin
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: function of one variable, large gradients, special formula for numerical differentiation, error estimate
Abstract
Numerical differentiation of functions with large gradients is investigated. It is assumed that a function contains a component known up to a factor and responsible for the large gradients of the function. Application of classical formulas for calculating derivatives to such functions may lead to significant errors. Special-purpose formulas are studied for numerical differentiation on a uniform grid which are exact for a boundary layer component. Conditions are formulated under which an error estimate of a difference formula for a derivative does not depend on the gradients of the boundary layer component. In the case of an exponential boundary layer, when calculating a derivative of an arbitrarily given order error estimates that are uniform with respect to a small parameter are obtained. The results of numerical experiments are presented.
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