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

In this paper we propose and analyze composite Filon--Clenshaw--Curtis quadrature rules for integrals of the form Iₖ[a,b](f,g) := ∫ₐᵇ f(x) exp(ikg(x)) d x, where 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 form CN k⁻ʳ M-N-1 + r, where r is determined by the strength of any singularity in f and the order of any stationary points in g and CN 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 Iₖ[a,b](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.