Gaussian quadrature rules for composite highly oscillatory integrals

TL;DR

This paper introduces two Gaussian quadrature rules tailored for highly oscillatory composite integrals, improving computational efficiency and accuracy, with theoretical analysis and numerical validation.

Contribution

The paper develops two novel Gaussian quadrature methods for composite oscillatory integrals, including one based on orthogonal polynomials and another for sign-changing functions, with convergence analysis.

Findings

The first rule's convergence depends on the integrand's regularity.

The second rule's nodes tend to the interval endpoints asymptotically.

Numerical experiments confirm the effectiveness of both methods.

Abstract

Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We show that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.