A numerical method for oscillatory integrals with coalescing saddle points

TL;DR

This paper develops a new numerical method using orthogonal polynomials to accurately evaluate highly oscillatory integrals with coalescing saddle points, overcoming issues faced by existing techniques.

Contribution

It introduces Gaussian quadrature rules based on complex orthogonal polynomials that remain accurate when saddle points coalesce, a classical problem in asymptotic analysis.

Findings

Constructed Gaussian quadrature rules for coalescing saddle points

Proved existence of orthogonal polynomials for even degrees

Developed an efficient numerical scheme for evaluation

Abstract

The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points.