A Levin method for logarithmically singular oscillatory integrals

TL;DR

The paper tackles the computation of oscillatory integrals with logarithmic singularities by extending the Levin method through a singularity-separation technique. It reformulates the problem into non-singular ODEs via $p(x)=q(x)\log x+h(x)$ and provides two algorithms: one for linear oscillators and one for general oscillators, both solvable by Chebyshev collocation without moment computations. The analysis proves equivalence to Filon for linear oscillators and establishes $\mathcal{O}(w^{-2}(1+\log|w|))$ convergence in frequency, with quasi-superalgebraic convergence for analytic $f$; numerical experiments confirm high accuracy and efficiency relative to Filon-based methods. The method yields stable, high-frequency-friendly performance for challenging integrals in physics and engineering and can be extended to other singularities or stationary points.

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.

Figures (1)

• Figure 1: Absolute errors scaled by $\frac{w^2}{1+\log|w|}$ for computing $\int_0^1e^{x}\log(x)e^{iwx}dx$ (denoted by Linear) and $\int_0^1(2x+1) e^{x^2+x}\log(x)e^{iw(x^2+x)}dx$ (denoted by Nonlinear) by the new Levin method for fixed $n$