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 I[a,b] k (f, g) := fkb a f(x) exp(ikg(x))dx, 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 CNk-rM-N-1+γ, 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 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.