Computing oscillatory integrals: Partition of the integration interval based on the singularity and the wave number of the integrand

Abstract

In this paper, we develop composite moment-free numerical quadratures for computing highly os- cillatory integrals with singularities and stationary points. The composite quadrature rules for computing highly oscillatory integrals with a smooth integrand and without a stationary point are developed based on partitioning the integration domain according to the values of the derivative of the oscillator and the wave number. The moment-free Filon-type quadrature is used for each of the oscillatory integrals de ned on the subintervals. The composite quadrature rules for computing highly oscillatory integrals with singularities and stationary points are developed by partitioning the integration domain according to the singularity of the integrand and the wave number, such that the integral de ned on a subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity or stationary point. The classical quadrature rules for weakly singular integrals using graded points are employed for computing the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for computing the oscillatory integrals. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator or to utilize the integral of special functions.