On the Conjecture of Stability Preservation in Arbitrary-Order Adams-Bashforth-Type Integrators

TL;DR

This paper analyzes the stability of a high-order explicit time stepping scheme, disproves a conjecture about its infinite accuracy stability, and provides criteria for maximum accuracy and stability in PDEs.

Contribution

It offers a rigorous harmonic analysis that disproves the conjecture and introduces a unified stability analysis strategy for high-order schemes.

Findings

Disproved the conjecture that the method remains stable at infinite accuracy.

Presented a criterion for maximum permissible accuracy based on stability radius.

Showed the method has enhanced stability compared to classical schemes.

Abstract

This paper presents stability and accuracy analysis of a high-order explicit time stepping scheme introduced by \cite[Section 2.2]{Buvoli2019}, which exhibits superior stability compared to classical Adams-Bashforth. A conjecture that is supported by several numerical phenomena in \cite[Figure 2.5]{Buvoli2018}, the method appears to remain stable when the accuracy approaches infinity, although it is not yet proven. We have disproven this conjecture from the perspective of harmonic analysis in this work. Notwithstanding the aforementioned, this method displays considerably enhanced stability in comparison to conventional explicit schemes. Furthermore, we present a criterion for ascertaining the maximum permissible accuracy for a given specific parabolic stability radius. Conversely, the original method will lose one order associated with the expected accuracy, which can be explored theoretically. Consequently, a unified analysis strategy for the \( L^2 \)-stability will be presented for extensional PDEs under the CFL condition. Finally, a selection of representative numerical examples will be shown in order to substantiate the theoretical analysis.