On the recent advances of spectral analysis for systems arising from fully-implicit RK methods

TL;DR

This paper unifies two spectral analysis approaches for matrices from fully implicit Runge-Kutta methods applied to linear PDEs, demonstrating their equivalence and consolidating existing results.

Contribution

It establishes the equivalence of different spectral analysis results and approaches for matrices from implicit RK methods, unifying previously separate methods.

Findings

Proves the equivalence of spectral analysis results from different formulations.

Unifies the approaches used in spectral analysis of implicit RK matrices.

Provides a comprehensive understanding of spectral properties in this context.

Abstract

This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same problem and used different tools to arrive at results that do not immediately coincide. We show the equivalence of the results as well as the equivalence of the approaches, unifying the two directions.