Nonlinear stability of direct quadrature methods for Volterra integral equations

An important topic in the numerical analysis of Volterra integral equations is the stability theory. The main results known in the literature have been obtained on linear test equations or, at least, on nonlinear equations with convolution kernel. Here, we consider Volterra integral equations with Hammerstein nonlinearity, not necessarily of convolution type, and we study the error equation for Direct Quadrature methods with respect to bounded perturbations. For a class of Direct Quadrature methods, we obtain conditions on the stepsize h for the numerical solution to behave stably and we report numerical examples which show the robustness of this nonlinear stability theory.