On the definition of the stability region of multistep methods

TL;DR

This paper critiques the traditional definition of stability regions for implicit multistep methods, highlighting the issue of isolated stable points that are often overlooked and proposing a refined definition to exclude them.

Contribution

It introduces a revised definition of stability regions that excludes isolated points, improving the accuracy of stability analysis for multistep methods.

Findings

Isolated stable points can occur when the leading coefficient vanishes.

Traditional root locus methods may miss these isolated points.

A new definition improves stability region characterization.

Abstract

The usual definition of the stability region of implicit multistep methods often implies that there are some isolated points of stability within the region of instability of the numerical method. These isolated stable points may appear when the leading coefficient of the characteristic polynomial of the method vanishes---they cannot be detected by the well-known root locus method, and their existence renders many results about stability regions problematic. It is suggested that the definition of the stability region should exclude such isolated points.