Filon–Clenshaw–Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities

In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of NN, the number of nodes, kk the rate of oscillations and a Sobolev-like regularity of the function. We prove that the method is not only robust but the error even decreases, for fixed NN, as kk increases. Practical issues about the implementation of the rule are also covered in this paper by: (a) writing down ready-to-implement algorithms; (b) analysing the numerical stability of the computations and (c) estimating the overall computational cost. We finish by showing some numerical experiments which illustrate the theoretical results presented in this paper.