On the relation between the repetition factor and numerical stability of direct quadrature methods for second kind Volterra integral equations

We consider direct quadrature methods employing quadrature rules which are reducible to linear multistep methods for ordinary differential equations. A simple characterization of both the repetition factor and numerical stability (for small $h$) is given, which enables us to derive some results with respect to a conjecture of Linz. In particular we show that (i) methods with a repetition factor of one are always numerically stable; (ii) methods with a repetition factor greater than one are not necessarily numerically unstable. Analogous results are derived with respect to the more general notion of an asymptotic repetition factor. We also discuss the concepts of strong stability, absolute stability and relative stability and their (dis)connection with the (asymptotic) repetition factor. Some numerical results are presented as a verification