An adaptive delaminating Levin method in two dimensions

TL;DR

This paper introduces an adaptive, resonance-agnostic Levin method for efficiently evaluating two-dimensional oscillatory integrals, achieving high accuracy even with stationary or resonance points present.

Contribution

It develops a novel adaptive delaminating Levin method that removes non-resonance restrictions and provides rigorous error estimates across all frequency regimes.

Findings

Method achieves high accuracy regardless of resonance conditions.

Adaptive subdivision improves efficiency in 2D oscillatory integral evaluation.

Numerical experiments confirm robustness and effectiveness.

Abstract

We present an adaptive delaminating Levin method for evaluating bivariate oscillatory integrals over rectangular domains. Whereas previous analyses of Levin methods impose non-resonance conditions that exclude stationary and resonance points, we rigorously establish the existence of a slowly-varying, approximate solution to the Levin PDE across all frequency regimes, even when the non-resonance condition is violated. This allows us to derive error estimates for the numerical solution of the Levin PDE via the Chebyshev spectral collocation method, and for the evaluation of the corresponding oscillatory integrals, showing that high accuracy can be achieved regardless of whether or not stationary and resonance points are present. We then present a Levin method incorporating adaptive subdivision in both two and one dimensions, as well as delaminating Chebyshev spectral collocation, which is effective in both the presence and absence of stationary and resonance points. We demonstrate the effectiveness of our algorithm with a number of numerical experiments.