Natural continuous extensions for Runge-Kutta methods for Volterra integrodifferential equations

In this paper we deal with a very general class of Runge-Kutta methods for the numerical solution of Volterra integro-differential equations. Our main contribution is the development of the theory of Natural Continuos Extensions (NCEs), i.e. piecewise polynomials functions which interpolate the values given by the Runge-Kutta methods at mesh points. The particular features of the NCEs allow to construct tail approximations which are quite efficient since they require a minimal number of kernel evaluations