A well-conditioned and efficient Levin method for highly oscillatory integrals with compactly supported radial basis functions

Abstract

Highly oscillatory integrals are frequently involved in applied problems, particularly for large-scale data and high frequencies. Levin method with global radial basis functions was implemented for numerical evaluation of these integrals in the literature. However, when the frequency is large or nodal points are increased, the Levin method with global radial basis functions faces several issues such as large condition number of the interpolation matrix and computationally inefficiency of the method, etc. In this paper, the Levin method with compactly supported radial basis functions is proposed to handle deficiencies of the method. In addition, theoretical error bounds and stability analysis of the proposed methods are performed. Several numerical examples are included to verify the accuracy, efficiency, and well-conditioned behavior of the proposed methods.