General linear methods for Volterra integral equations

Abstract

We investigate the class of general linear methods of order $p$ and stage order $q=p$ for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given

On the numerical stability of the linear barycentric rational quadrature method for Volterra integral equations

“…The CBRQM (9) is said to be stable with respect to (17) for a given (x, y) ∈ R 2 if, for that (x, y), all the roots of Φ(w, x, y) = 0, defined by (20), lie in the unit disk with only simple roots on the boundary.…”

“…Following [13,20] we can define the region of stability of the CBRQM (9) with respect to the convolution test equation (17) which is related to the stability function Φ(w, x, y).…”

“…A wide variety of numerical methods have been presented to solve this kind of equation (e.g., [1,9,11,12,15,18,19,20]). An elegant, accurate and efficient familiar class of the proposed methods is based on a direct quadrature rule.…”

Construction of efficient general linear methods for stiff Volterra integral equations

“…In this regard, A-and V 0 -stable RungeKutta methods for VIEs have been introduced in [1,5,11]. Like general linear methods (GLMs) which have been introduced as a unifying framework for the traditional methods for solving initial value problems [3,12], Izzo et al [10] investigated the class of GLMs of order p and stage order q = p for the numerical solution of (1) to analyze stability properties of the method conveniently. GLMs combine the essential multivalue and multistage natures of the methods.…”

“…In the introduced GLMs in [10], it is assumed that the input and output vector have the Nordsieck form. In this paper we consider GLMs for the numerical solution of (1) in general form, rather than Nordsieck form.…”