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Computing highly oscillatory integrals

Authors: Yunyun Ma and Yuesheng Xu
Journal: Math. Comp. 87 (2018), 309-345
MSC (2010): Primary 65D30, 65D32
Published electronically: April 7, 2017
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Abstract: We develop two classes of composite moment-free numerical quadratures for computing highly oscillatory integrals having integrable singularities and stationary points. One class of the quadrature rules has a polynomial order of convergence and the other class has an exponential order of convergence. We first modify the moment-free Filon-type method for the oscillatory integrals without a singularity or a stationary point to accelerate its convergence. We then consider the oscillatory integrals without a singularity or a stationary point and then those with singularities and stationary points. The composite quadrature rules are developed based on partitioning the integration domain according to the wave number and the singularity of the integrand. The integral defined on the resulting subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity. Classical quadrature rules for weakly singular integrals using graded points are employed for the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for the oscillatory integrals without a singularity. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator which normally is a nontrivial task. Numerical experiments are presented to demonstrate the approximation accuracy and the computational efficiency of the proposed methods. Numerical results show that the proposed methods outperform methods published recently.

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Additional Information

Yunyun Ma
Affiliation: School of Data and Computer Science, and Guangdong Province Key Lab of Computational Science, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China

Yuesheng Xu
Affiliation: School of Data and Computer Science, and Guangdong Province Key Lab of Computational Science, Sun Yat-sen University, Guangzhou 510275, People’s Republic of China

Keywords: Oscillatory integrals, algebraic singularities, stationary points, moment-free Filon-type method, graded points
Received by editor(s): August 3, 2014
Received by editor(s) in revised form: November 15, 2015, and July 27, 2016
Published electronically: April 7, 2017
Additional Notes: This research was supported in part by the Ministry of Science and Technology of China under grant 2016YFB0200602, by the US National Science Foundation under grant DMS-1522332, by Guangdong Provincial Government of China through the Computational Science Innovative Research Team program and by the Natural Science Foundation of China under grants 11471013 and 91530117
The second author is also a Professor Emeritus of Mathematics, Syracuse University, Syracuse, New York 13244
Article copyright: © Copyright 2017 American Mathematical Society

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