Abstract: Numerical methods for approximating very oscillatory integrals have gained a renewed interest in the last years. For instance, in many numerical methods for solving scattering problems in the high frequency regime, there appear integrals whose oscillations are proportional, at best, to the wave number of the incident wave. The very well-known rule of thumb of using 10-20 points per oscillation to capture the integrals properly leads to prohibitively expensive methods, even in the simpler problems. Furthermore the integrand itself can be singular, which has to be considered too.
In this talk we present a Filon-Clenshaw-Curtis approach for approximating such integrals. The analysis we present for these rules is complete: We derive convergence rates, in terms of the number of nodes and the "strength" of the oscillations, we show an efficient way to implement the resulting algorithms, and we prove their numerical stability. The use of graded meshes, needed to take care of any singularity in the integrand, is also explored. We show that, with the right choice of the grading parameters, the original good convergence rates can be restored in this case too.
This is joint work with Tatiana Kim and Ivan G. Graham (University of Bath, UK) and V.P. Smyshlyaev (University College London, UK).