On the adaptive Levin method

TL;DR

This paper proves that the adaptive Levin method, when discretized with a Chebyshev spectral approach and solved via SVD, does not suffer from low-frequency breakdown, even near stationary points, enhancing the evaluation of oscillatory integrals.

Contribution

The paper provides a rigorous proof that the adaptive Levin method remains accurate for slowly oscillating integrals and stationary points, supporting its theoretical foundation.

Findings

No low-frequency breakdown occurs with the spectral discretization and SVD approach.

The adaptive Levin method effectively handles stationary points in oscillatory integrals.

Numerical experiments confirm the theoretical results and demonstrate the method's robustness.

Abstract

The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this approach suffers from "low-frequency breakdown," meaning that the accuracy of the calculated value of the integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence, however, suggests that if a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, then no low-frequency breakdown occurs. Here, we provide a proof that this is the case, and our proof applies not only when the integrand is slowly oscillating, but even in the case of stationary points. Our result puts adaptive schemes based on the Levin method on a firm theoretical foundation and accounts for their behavior in the presence of stationary points. We go on to point out that by combining an adaptive Levin scheme with phase function methods for ordinary differential equations, a large class of oscillatory integrals involving special functions, including products of such functions and the compositions of such functions with slowly-varying functions, can be easily evaluated without the need for symbolic computations. Finally, we present the results of numerical experiments which illustrate the consequences of our analysis and demonstrate the properties of the adaptive Levin method.