A Levin method for logarithmically singular oscillatory integrals

TL;DR

This paper introduces a stable Levin method for efficiently computing oscillatory integrals with logarithmic singularities, transforming the problem into non-singular ODEs for improved accuracy and convergence.

Contribution

A novel Levin method that handles logarithmic singularities by singularity separation and transforms the problem into non-singular ODEs, enhancing stability and efficiency.

Findings

Method effectively computes singular oscillatory integrals

Demonstrates higher accuracy for high oscillation frequencies

Numerical experiments validate efficiency and stability

Abstract

We propose a new stable Levin method to compute oscillatory integrals with logarithmic singularities and without stationary points. To avoid the singularity, we apply the technique of singularity separation and transform the singular ODE into two non-singular ODEs, which can be solved efficiently by the collocation method. Applying the equivalency of the new Levin method for the singular oscillatory integrals and the Filon method when the oscillator is linear, we consider the convergence of the new Levin method. This new method shares the proposition that less error for higher oscillation. Several numerical experiments are presented to validate the efficiency of the proposed method.