Approximation properties of quadrature methods for Volterra integral equations of the first kind

Abstract

We present a unifying analysis of quadrature methods for Volterra integral equations of the first kind that are zero-stable and have an asymptotic repetition factor. We show that such methods are essentially collocation-projection methods with underlying subspaces that have nice approximation properties, and which are stable as projection methods. This is used to derive asymptotically optimal error estimates under minimal smoothness conditions. The class of quadrature methods covered includes the cyclic linear multistep and the reducible quadrature methods, but not (really) Runge-Kutta methods.