Finite Difference Weights, Spectral Differentiation, and Superconvergence

TL;DR

This paper presents an improved algorithm for computing finite difference weights that is more efficient and accurate, extends to spectral differentiation, and explores superconvergence phenomena in finite difference formulas.

Contribution

It introduces a faster, equally accurate algorithm for finite difference weights, generalizes to spectral differentiation, and analyzes superconvergence in finite difference formulas.

Findings

The new algorithm reduces arithmetic operations by a factor of 4/(5m+5).

Finite difference weights can be computed accurately for derivatives up to order 16.

Superconvergence can occur, but the order of accuracy cannot be boosted by more than 1 for real grid points.

Abstract

Let $z_{1},z_{2},...,z_{N}$ be a sequence of distinct grid points. A finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as $\sum w_{k}f(z_{k})$, with $w_{k}$ being the weights. We derive an algorithm for finding the weights $w_{k}$ which is an improvement of an algorithm of Fornberg (\emph{Mathematics of Computation}, vol. 51 (1988), p. 699-706). This algorithm uses fewer arithmetic operations than that of Fornberg by a factor of $4/(5m+5)$ while being equally accurate. The algorithm that we derive computes finite difference weights accurately even when $m$, the order of the derivative, is as high as 16. In addition, the algorithm generalizes easily to the efficient computation of spectral differentiation matrices. The order of accuracy of the finite difference formula for $f^{(m)}(0)$ with grid points $hz_{k}$, $1\leq k\leq N$, is typically $\mathcal{O}(h^{N-m})$. However, the most commonly used finite difference formulas have an order of accuracy that is higher than the typical. For instance, the centered difference approximation $(f(h)-2f(0)+f(-h))/h^{2}$ to $f"(0)$ has an order of accuracy equal to 2 not 1. Even unsymmetric finite difference formulas can exhibit such superconvergence or boosted order of accuracy, as shown by the explicit algebraic condition that we derive. If the grid points are real, we prove a basic result stating that the order of accuracy can never be boosted by more than 1.