Efficient algorithms for integrals with highly oscillatory Hankel kernels

Abstract

We present an efficient numerical algorithm for approximating integrals with highly oscillatory Hankel kernels. First, using the integral expression of the Hankel function, the integral is transformed into the integral of the trigonometric kernel, and the singularities are separated. According to whether the integration interval contains zeros or not, the integrals are divided into two cases: singular highly oscillatory integrals and non-singular highly oscillatory integrals. For the non-singular case, the Clenshaw–Curtis–Filon rules (CCF) method is used for fast calculation. For the second case, by using the Cauchy integral theorem, the integral is transformed into an infinite integral, which can be calculated by generalized Gaussian–Laguerre rules. The efficiency and accuracy of the new method are verified by numerical examples.

Keywords

• Hankel integral
• Highly oscillatory integrals
• Clenshaw–Curtis methods
• Generalized Gauss–Laguerre rule
• Singular integral