Stability of numerical approximations to Volterra integral equations

In the mathematical representation of real life history-dependent problems (such as mechanical systems, electric circuit, epidemiology, population growth,...), systems of Volterra integral equations are widely used. A reliable numerical simulation of these phenomena requires a careful analysis of the long time behavior of the numerical solution. This analysis is usually carried out on model problems which are chosen as close as possible to the applications. Here we develop a numerical stability theory for Direct Quadrature (DQ) methods which applies to a quite general and representative class of problems. We obtain stability results under some conditions on the stepsize and, in particular cases, unconditional stability for DQ methods of whatever order.