Construction of efficient general linear methods for stiff Volterra integral equations

Abstract

General linear methods in the Nordsieck form have been introduced for the numerical solution of Volterra integral equations. In this paper, we introduce general linear methods of order p and stage order q = p for the numerical solution of Volterra integral equations in general form, rather than Nordsieck form. A-and V 0 (α)-stable methods are constructed and applied on stiff problems to show their efficiency.

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

“…The construction of methods with unbounded stability regions with respect to (11) and (17) (especially A-and V 0 -stable methods) in this class is still an open problem which will be attempted in future work. We recall that A. Abdi (the second author) et al have introduced A-and V 0 (α)-stable methods, with maximum α, in the other class of discritization methods in [1].…”

“…Theorem 1. Let n and d, d ≤ n, be positive integers, let the nodes be equispaced, and let f ∈ C d+2 (I) and K ∈ C d+2 (Ω) in (1). Consider the approximate solution of (1) by (10) and the CBRQM (9).…”

“…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.…”

General linear methods with large stability regions for Volterra integral equations

“…Izzo et al (2010) introduced GLMs for the numerical solution of VIEs (1). They constructed Nordsieck GLMs of order p = 1, 2 and stage order q = p in which input and output vectors of the methods approximate the Nordsieck vector of order p. Abdi et al (2016) investigated the class of GLMs for the numerical solution of (1) in general form, rather than Nordsieck form. They constructed methods of order p and stage order q = p up to four.…”

“…for some real numbers Abdi et al 2016). In an GLM for (1), these quantities are related by the formulas (Abdi et al 2016;Izzo et al 2010)…”