Levin methods for highly oscillatory integrals with singularities

TL;DR

This paper introduces advanced Levin methods for efficiently computing highly oscillatory integrals with singularities, utilizing singularity separation and non-singular ODEs to achieve high accuracy and convergence.

Contribution

The paper develops new Levin methods that handle singularities in oscillatory integrals by converting singular ODEs into non-singular forms, enabling high-order and superalgebraic convergence.

Findings

Methods achieve high asymptotic order

Numerical experiments validate efficiency

Superalgebraic convergence demonstrated

Abstract

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.