Error Analysis of Finite Differences and the Mapping Parameter in Spectral Differentiation

TL;DR

This paper investigates the errors in spectral differentiation using Chebyshev points, critiques existing mapping parameter choices, and proposes improved methods for accuracy enhancement.

Contribution

It provides a comprehensive analysis of rounding and discretization errors, offering a more complete justification for selecting the mapping parameter in spectral differentiation.

Findings

Rounding error increases faster than expected with derivative order.

The original mapping parameter choice is incomplete and can be improved.

Using the discrete cosine transform allows for a better mapping parameter choice.

Abstract

The Chebyshev points are commonly used for spectral differentiation in non-periodic domains. The rounding error in the Chebyshev approximation to the $n$-the derivative increases at a rate greater than $n^{2m}$ for the $m$-th derivative. The mapping technique of Kosloff and Tal-Ezer (\emph{J. Comp. Physics}, vol. 104 (1993), p. 457-469) ameliorates this increase in rounding error. We show that the argument used to justify the choice of the mapping parameter is substantially incomplete. We analyze rounding error as well as discretization error and give a more complete argument for the choice of the mapping parameter. If the discrete cosine transform is used to compute derivatives, we show that a different choice of the mapping parameter yields greater accuracy.