Spijker's example and its extension

TL;DR

This paper investigates the stability differences between strongly and weakly stable linear multistep methods, extending Spijker's example to better understand the properties that distinguish their stability behaviors.

Contribution

The paper extends Spijker's example to the general weakly stable case, providing insights into the stability properties of linear multistep methods.

Findings

Weakly stable methods can produce spurious oscillations.

Spijker's example demonstrates instability in the Spijker norm.

Extension of Spijker's result to general weakly stable methods.

Abstract

Strongly and weakly stable linear multistep methods can behave very differently. The latter class can produce spurious oscillations in some of the cases for which the former class works flawlessly. The main question is if we can find a well defined property which clearly tells the difference between them. There are many explanations from different viewpoints. We cite Spijker's example which shows that the explicit two step midpoint method is unstable with respect to the Spijker norm. We show that this result can be extended for the general weakly stable case.