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-k((f, g))[a, b] := f(a)(b) f(x)epx(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 C(N)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 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 -> 8 is the same as would be obtained if f was smooth. Moreover, the quadrature error decays at least as fast as k -> 8 as does the original integral I-k([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.