Fast and stable augmented Levin methods for highly oscillatory and singular integrals

Wang, Yinkun; Xiang, Shuhuang

doi:10.1090/mcom/3725

Fast and stable augmented Levin methods for highly oscillatory and singular integrals
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by Yinkun Wang and Shuhuang Xiang;

Math. Comp. 91 (2022), 1893-1923

DOI: https://doi.org/10.1090/mcom/3725

Published electronically: February 15, 2022

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Abstract:

In this paper, augmented Levin methods are proposed for the computation of oscillatory integrals with stationary points and an algebraically or logarithmically singular kernel. Different from the conventional Levin method, to overcome the difficulties caused by singular and stationary points, the original Levin ordinary differential equation (Levin-ODE) is converted into an augmented ODE system, which can be fast and stably implemented with a cost of $O(n\log n$) by applying sparse and fast spectral methods together with the truncated singular value decomposition. The established asymptotics and convergence show that these schemes become more accurate as the frequency increases and are super-algebraically convergent. The effectiveness and accuracy were tested by numerical examples, showing perfect coincidence with the estimates.

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