Title: Simple and compact finite difference formulae using real and complex variables

Authors: Yohei Nishidate

Addresses: The University of Aizu, Aizu-Wakamatsu, Fukushima, 9658580, Japan

Abstract: A new set of compact finite difference formulae is derived by simple combinations of the real and the complex Taylor series expansions. The truncation error is fourth-order in derived formulae for approximating first to fourth-order derivatives. Although there exist complex stencil finite difference formulae with better truncation errors, our formulae are computationally cheaper, requiring only three points for first to third-order and four points for fourth-order derivatives. The derived formulae are experimented with for approximating derivatives of relatively simple and highly nonlinear functions used in other literature. Although the new formulae suffer the subtractive cancellation, it is demonstrated that the derived formulae outperform finite difference formulae of comparable computational costs for relatively large step sizes.

Keywords: Taylor series expansion; approximation in the complex domain; finite difference methods; compact finite difference formula; numerical approximation.

DOI: 10.1504/IJCSE.2023.135283

International Journal of Computational Science and Engineering, 2023 Vol.26 No.6, pp.684 - 693

Received: 01 May 2022
Received in revised form: 05 Aug 2022
Accepted: 30 Aug 2022
Published online: 04 Dec 2023 *

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