Efficient BBFM-collocation for weakly singular oscillatory Volterra integral equations of the second kind

We present a fast and accurate numerical scheme for approximating weakly singular oscillatory Volterra integral equations (VIEs) of the second kind. The main idea is the combination of the black-box fast multipole method (FMM) and collocation methods that leads to a significant speed-up in CPU time for a given tolerance. The weakly singular oscillatory integrals, which arise from the evaluation of integral equations, are calculated using Clenshaw-Curtis Filon methods. Because the kernel function is approximated by a Chebyshev interpolation scheme, it is very useful for the kernel which is known analytically but is quite complicated. Compared with the method and the error estimates reported previously, the new scheme is suitable for more general kernel function and provide sharper estimates. The efficiency and accuracy of the new method are verified by numerical examples.