Numerical integrating of highly oscillating functions: effective stable algorithms in case of linear phase

TL;DR

This paper introduces a stable, efficient numerical method for calculating Fourier integrals of highly oscillating functions with linear phase, applicable at various frequencies, using collocation and Chebyshev differentiation matrices.

Contribution

The paper presents a novel stable algorithm combining collocation with Chebyshev differentiation matrices for efficient Fourier integral computation of oscillating functions.

Findings

Effective at low and high frequencies

Reduces integral calculation to solving banded linear systems

Improves numerical stability and efficiency

Abstract

A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly oscillating part of the antiderivative of the desired integral, allows reducing the calculation of the integral of a highly oscillating function (with a linear phase) to solving a system of linear algebraic equations with a three-diagonal triangular or five-diagonal band Hermitian matrix. The choice of Gauss-Lobatto grid nodes as collocation points makes it possible to use the properties of discrete "orthogonality" of Chebyshev differentiation matrices in physical and spectral spaces. This is realized by increasing the efficiency of the numerical algorithm for solving the problem. The system pre-conditioning procedure leads to significantly less cumbersome and more economical calculation formulas. To avoid possible numerical instability of the algorithm, we proceed to the solution of a normal system of linear algebraic equations.