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In this paper we propose and analyse composite Filon-Clenshaw-Curtisquadrature rules for integrals of the form $I_{k}^{[a,b]}(f,g) := \int_a^b f(x)\exp(\mathrm{i}kg(x)) \rd x $, where $k \geq 0$, $f$ may have integrablesingularities and $g$ may have stationary points. Our composite rule is definedon a mesh with $M$ subintervals and requires $MN+1$ evaluations of $f$. Itsatisfies an error estimate of the form $C_N k^{-r} M^{-N-1 + r}$, where $r$ isdetermined by the strength of any singularity in $f$ and the order of anystationary 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 $g$ areexplicit in the error estimates. For fixed $k$, the rate of convergence of therule as $M \rightarrow \infty$ is the same as would be obtained if $f$ wassmooth. Moreover, the quadrature error decays at least as fast as $k\rightarrow \infty$ as does the original integral $I_{k}^{[a,b]}(f,g)$. For thecase of nonlinear oscillators $g$, the algorithm requires the evaluation of$g^{-1}$ at non-stationary points. Numerical results demonstrate the sharpnessof the theory. An application to the implementation of boundary integralmethods for the high-frequency Helmholtz equation is given.

Filon-Clenshaw-Curtis rules for highly-oscillatory integrals with algebraic singularities and stationary points