Geodesic
Formulas for numerical differentiation of functions with large gradients
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 1, pp. 17-26

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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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     author = {A. I. Zadorin},
     title = {Formulas for numerical differentiation of functions with large gradients},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     publisher = {mathdoc},
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     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2023_26_1_a1/}
}
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A. I. Zadorin. Formulas for numerical differentiation of functions with large gradients. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 1, pp. 17-26. http://geodesic.mathdoc.fr/item/SJVM_2023_26_1_a1/