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

2015/10/31 by Divakar Viswanath, Viswanath, Divakar
Computer Science · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1511.00137

openalex publication_date 2015/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (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.

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