Báo Cáo Báo cáo hóa học ONE PARAMETER FAMILY OF LINEAR DIFFERENCE EQUATIONS AND THE STABILITY PROBLEM FOR TH

ONE PARAMETER FAMILY OF LINEAR DIFFERENCE EQUATIONS AND THE STABILITY PROBLEM FOR THE NUMERICAL SOLUTION OF ODEs L. ACETO, R. PANDOLFI, AND D. TRIGIANTE Received 21 July 2004; Accepted 4 October 2004

The study of the stability properties of numerical methods leads to considering linear difference equations depending on a complex parameter q. Essentially, the associated characteristic polynomial must have constant type for q ∈ Cư . Usually such request is proved with the help of computers. In this paper, by using the fact that the associated polynomials are solutions of a “Legendre-type” difference equation, a complete analysis is carried out for the class of linear multistep methods having the highest possible order. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

The problem to approximate the solutions of differential equations by substituting to them “appropriate” difference equations is as old as the differential calculus.