Test equations and linear stability of implicit-explicit general linear methods

TL;DR

This paper applies eigenvalue perturbation theory to analyze the stability of implicit-explicit general linear methods (IMEX GLMs) for linear ODEs, including complex-valued equations with parabolic and hyperbolic stiffness, supported by practical stability analyses.

Contribution

It introduces a novel eigenvalue perturbation approach to characterize IMEX GLM stability without assuming matrix diagonalizability or symmetry, extending stability analysis to complex-valued equations.

Findings

Eigenvalue perturbation theory justifies test equations for stability analysis.

Stability results apply to complex-valued scalar ODEs with parabolic and hyperbolic stiffness.

Practical stability analysis demonstrated on shallow-water and advection-diffusion models.

Abstract

Eigenvalue perturbation theory is applied to justify using complex-valued linear scalar test equations to characterize the stability of implicit-explicit general linear methods (IMEX GLMs) solving autonomous linear ordinary differential equations (ODEs) when the implicitly treated term is sufficiently stiff relative to the explicitly treated term. The stiff and non-stiff matrices are not assumed to be simultaneously diagonalizable or triangularizable and neither matrix is assumed to be symmetric or negative definite. The stability of IMEX GLMs solving complex-valued scalar linear ODEs displaying parabolic and hyperbolic stiffness is analyzed and related to the higher dimensional theory. The utility of the theoretical results is highlighted with a stability analysis of a family of IMEX Runge-Kutta methods solving IVPs of a linear 2D shallow-water model and a linear 1D advection-diffusion model.