On two-step continuous methods for Volterra Integral Equations

The aim of this talk is to present highly stable collocation based numerical methods for Volterra Integral Equations (VIEs). As it is well known, a collocation method is based on the idea of approximating the exact solution of a given integral equation with a suitable function belonging to a chosen finite dimensional space, usually a piecewise algebraic polynomial, which satisfies the integral equation exactly on a certain subset of the integration interval (called the set of collocation points). The collocation technique allows the derivation of methods having many desirable properties. In fact, collocation methods provide an approximation at each point of the integration interval to the solution of the equation, thus leading, in general, to a cheap variable stepsize implementation. Moreover, the collocation function can be expressed as a linear combination of functions ad hoc for the problem we are integrating, in order to better reproduce the qualitative behaviour of the solution. Two-step collocation and modified collocation methods have already been developed in the context of Ordinary Differential Equations [6,7] and VIEs [3-5] with the aim of increasing the order of classical one-step methods, without any increase of the computational cost, maintaining desirable stability properties. In particular we aim to present new results obtained in the context of twostep almost collocation methods for VIEs, i.e. methods obtained by relaxing some collocation and/or interpolation conditions in order to obtain high uniform order of convergence together with strong stability properties. We will exploit the continuous order conditions in order to provide a possible error estimation which will be at the basis of a variable stepsize implementation and show numerical experiments that confirm the theoretical expectations.