Discretized multistep collocation methods for Volterra integro-differential equations

Volterra Integro-Differential Equations (VIDEs) have been proposed as the mathematical models of a wide class of evolutionary problems with memory, such as viscoelastic materials with memory, in which the stress at each point depends both on the present value of the deformation gradient and on the entire temporal history of motion. In this talk we present a promising numerical approach based on multistep collocation. Recently have been introduced exact multistep collocation methods which approximate the solution by a piecewise algebraic polynomial, which satisfies the VIDE on the collocation points. The collocation polynomial depends on a fixed number of previous time-steps (instead of only one, as in the classical one-step methods). These methods have higher order of convergence at the same computational cost of one-step collocation ones, have strong stability properties, and provide an approximation of the solution at each point of the time interval, that is quite useful in a variable step implementation. On the other side, they cannot be directly implemented since they require the evaluation of some integrals. Here we describe the construction of discretized multistep collocation methods, where the collocation polynomial is derived by using suitable quadrature formulas for the approximation of such integrals. We analyze the convergence and the numerical stability of the proposed methods. In particular we establish how to choose quadrature rules which preserve the order of convergence of exact methods, find classes of A0-stable methods and derive an estimate of the local error. We show the performances of our methods on some significative examples.