Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains

Zhen Yang and Junjie Ma
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Zhen Yang: School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China
Junjie Ma: School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China

Mathematics, 2020, vol. 8, issue 11, 1-21

Abstract: In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

Keywords: highly oscillatory integral; Chebyshev polynomial; nearly singular; Levin quadrature rule; adaptive mesh refinement (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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