An Exploration of the Approximation of Derivative Functions via Finite Differences

TL;DR

This paper investigates the limitations of finite difference methods for derivative approximation on nonuniform grids, highlighting issues of accuracy and consistency through analysis and simulations.

Contribution

It provides a detailed analysis of the accuracy and consistency problems of finite difference formulas on nonuniform grids, which is less explored in existing literature.

Findings

Finite differences lose accuracy on nonuniform grids.

Difference formulas for second derivatives can be inconsistent.

Simulations demonstrate the impact of grid irregularity on approximation quality.

Abstract

Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent approximations to various derivative functions, including those used in modeling important physical processes on uniform grids. However, our research reveals that difference approximations on uniform grids cannot be applied blindly on nonuniform grids, nor can difference formulas to form consistent approximations to second derivatives. At best, they may lose accuracy; at worst they are inconsistent. Detailed consistency and error analysis, together with simulated examples, will be given.