A Gaussian quadrature rule for Fourier-type highly oscillatory integrals in the presence of stationary points

Abstract

In this article, a Gaussian quadrature rule was developed for the approximation of Fourier-type highly oscillatory integrals (FHOIs) with the phase functions of form g(x)=xr,r=2s,s≥1,x∈[−1,1]. The presence of these rules was demonstrated numerically through an analytical-numerical process, and questions were also addressed regarding the number of the nodes of the quadrature rule tend to the endpoints ±1 and the number of those tend to the so-called stationary points (where the integrand does not oscillate locally) as ω→∞. With proper assumptions, an asymptotic order was obtained for the numerical method, which indicated that the quadrature error decreased rapidly with the increase in oscillation parameter ω or the quadrature points. Furthermore, the behavior of the quadrature nodes as ω→∞ and ω→0 was analyzed to demonstrate the optimal asymptotic order and optimal polynomial order of the proposed method. To illustrate the efficiency and accuracy of the proposed method, some numerical examples were considered as well. In addition, the proposed method was compared to other numerical methods.