Error Analysis of Finite Differences and the Mapping Parameter in Spectral Differentiation
This work refines a known numerical method for spectral differentiation, offering improved accuracy for practitioners in computational science.
The paper identifies an incomplete justification for the mapping parameter in Kosloff and Tal-Ezer's spectral differentiation technique and provides a more complete error analysis, showing that a different parameter choice yields greater accuracy when using the discrete cosine transform.
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.