Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals

Abstract

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form ∫ 1 −1 f (s) exp(iks) ds. We use a Filon approach based on interpolating f at the classical Clenshaw-Curtis points cos(jπ/N), j = 0,. .. , N. The rule may be implemented in O(N log N) operations. We prove error estimates which show explicitly how the error depends both on the parameters k and N and on the Sobolev regularity of f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. We also describe a method for implementating the method and prove its stability both when N ≤ k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

“…In the high-frequency case, the resulting integrals will be highly oscillatory. The efficient calculation of these type of integrals is an active area of research (see [10,29,36] and the references therein).…”

Error analysis of the extended Filon-type method for highly oscillatory integrals

“…For example, we have exactly six methods with d = 7: the pairs (s, ν) in {(1, 6), (1,5), (2,4), (2,3), (3,2), (3, 1)}. Note that we have two such methods for each s, for some odd ν s and for ν s + 1: the latter, according to our observations, has a smaller constant.…”

“…2. Another choice of internal points has been proposed in [5] for the case s = 1 and will be extended by us to all s ≥ 1, namely the Chebyshev points of the second kind cos kπ/(ν + 1), k = 1, . .…”

“…Insofar as interior points, inclusive of Jacobi points and Chebyshev points of the second kind, are not new and have been justified partly [5,13], in particular for the case ν 1. The purpose of this paper is a rigorous and complete analysis of EFM and the establishment of realistic and tight upper bounds on its numerical error.…”

“…1 two choices: EFJ, whereby the c k s are the zeros of the Jacobi polynomial P (s,s) ν , a choice that maximised the conventional order of the (nonoscillatory) quadrature for ω = 0 [11], and EFCC, with c k = cos(kπ/(ν + 1)) (incidentally, these are also zeros of a Jacobi polynomial, P ). The advantage of EFCC is that they are uniformly bounded [5]. 1 We commence by calculating r(1) = ν k=1 (1 − c k ) and r(−1) = (−1) ν ν k=1 (1 + c k ) for the two above configurations of inner points.…”