A Numerical Method for Oscillatory Integrals with Coalescing Saddle Points

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.

Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis

“…After extra transformations, we can also use Gauss-Laguerre quadrature. The integrals I ±1 (c) that follow from (3.32) can be computed in the same way, as is also proposed in [15]. We can do the integral I −1,1 (c)t with our approach, as follows from a simple transformation from (1.6) to (1.7).…”

“…In a recent paper [15], the approach also avoids developing an asymptotic expansion and computing the coefficients of the Airy-type asymptotic expansion and of the Airy functions. These authors use also numerical computation of the integral in the representation given in (1.1).…”

“…Their novel idea is based on Gauss quadrature on the complex contour C with polynomials that are orthogonal on this contour. This requires the computation of the zeros of the polynomials and moments, and these topics are discussed in detail in [15]; an earlier paper on this topic is [6]. In this way a uniform method of computation is obtained valid for general and certainly small values of the parameter η in (1.1).…”

“…In [15] the goal of the paper is the construction and analysis of a uniformly applicable quadrature rule, uniform in the parameter c near c = 0, for the canonical integral The parameters k and ω are large and when k ∼ ω the integral I 1 (k, ω) has two nearby saddle points (we use µ = k/ω)…”

“…In this way a uniform method of computation is obtained valid for general and certainly small values of the parameter η in (1.1). In [15] an asymptotic estimate is given for error terms in their Gauss quadrature approach. We see in that paper, once the nodes and weights of the quadrature rule have been made available, very good performances of the quadrature rule for rather small degree of the underlying orthogonal polynomials.…”