### Abstract

The approximation of solutions to Volterra integral equations by collocation or discontinuous Galerkin methods leads to a set of `semi-discretised' equations that in general are not amenable to numerical computation: an additional discretisation process that is able efficiently and accurately to cope with the highly oscillatory nature of the kernel of the given Volterra integral operator is needed. Here, the use of Filon-type quadrature is an obvious possibility; however, it is not yet clear how best to do this when the kernel is weakly singular.

In this talk I will describe current work related to the above problem, and especially to collocation methods for various types of Volterra functional equations, including equations with variable (and possibly vanishing) delay arguments. It will also be shown that in the case of Volterra integral equations of the first kind, the choice of the quadrature scheme in discontinuous Galerkin methods can have a major effect on the convergence properties of the approximate solution.