Computing highly oscillatory integrals

Ma, Yunyun; Xu, Yuesheng

doi:10.1090/mcom/3214

Computing highly oscillatory integrals
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by Yunyun Ma and Yuesheng Xu PDF

Math. Comp. 87 (2018), 309-345 Request permission

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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