Filon--Clenshaw--Curtis Rules for Highly Oscillatory Integrals with Algebraic Singularities and Stationary Points

Abstract

In this paper we propose and analyze composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form Iwhere k ≥ 0, f may have integrable singularities, and g may have stationary points. Our composite rule is defined on a mesh with M subintervals and requires MN + 1 evaluations of f . It satisfies an error estimate of the formwhere r is determined by the strength of any singularity in f and the order of any stationary points in g and C N is a constant which is independent of k and M but depends on N .The regularity requirements on f and g are explicit in the error estimates. For fixed k, the rate of convergence of the rule as M → ∞ is the same as would be obtained if f was smooth. Moreover, the quadrature error decays at least as fast as k → ∞ as does the original integral I [a,b] k (f, g). For the case of nonlinear oscillators g, the algorithm requires the evaluation of g −1 at nonstationary points. Numerical results demonstrate the sharpness of the theory. An application to the implementation of boundary integral methods for the high-frequency Helmholtz equation is given.

Computing highly oscillatory integrals

“…In the following theorem, we analyze the error E 7) and N Q [a,b] N,κ,m [f, g] ≤ N m + 1. In particular, if N is chosen as ⌈σ⌉, the smallest integer not less than σ, then…”

“…Numerical experiments are presented in Section 6 to confirm the theoretical estimates on the accuracy of the proposed formulas. Moreover, we compare the numerical performance of the proposed formulas with that of those recently proposed in [6,7]. We summarize our conclusions in Section 7.…”

“…The Filon-Clenshaw-Curtis rules [7] for the oscillatory integrals with nonlinear oscillator requires the evaluation of g −1 . Quadrature methods developed in this section do not require computing derivatives of f or those of g ′ , nor evaluating g −1 .…”

“…For the cases having nonlinear oscillators, it is not always possible to compute the moments exactly. In [6,7,27,37], the moment-free Filon-type methods were developed. An entirely different approach without computing the moments is the Levin collocation method [21].…”

“…In particular, certain special functions were used for calculating the oscillatory integrals in the case when f has singularities and g has stationary points. The formulas proposed in [6,7] do not require computing the special functions and the moments of these formulas can be computed exactly, at the expenses of computing the inverse of the oscillator, which takes up much computing time. Numerical steepest descent methods also require computing the inverse of the oscillator or high order derivatives of the integrand [2].…”