On the discretization of differential and Volterra integral equations with variable delay

In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss (-Legendre) points will no longer yield the optimal (local) orderO(h ^2m ). This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406).