Natural Continuous Extensions of Runge-Kutta methods for Volterra integrodifferential equations

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