On the Coefficients in Finite Difference Series Expansions of Derivatives

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2009-2025, October 2025.
Abstract. The formulation of finite difference approximations is a classical problem in numerical analysis. In this article, we consider difference approximations that are based on a series expansion in powers of the second undivided difference. Each additional term in the series increases the order of accuracy by two. These expansions are useful in a variety of contexts such as in the development of modified equation schemes, the design of high-order accurate energy stable discretizations, and error analysis of certain finite element or finite difference schemes. Here, we provide closed form expressions for the coefficients in the series expansions for derivatives of all orders. We also provide some short recursions defining the series coefficients, and formulae for the stencil coefficients in standard difference approximations. The series expansions are used to show some useful properties of the Fourier symbols of difference approximations and to derive rules of thumb for the number of points-per-wavelength needed to achieve a given error tolerance when solving wave propagation problems involving higher spatial derivatives.