Implicit-explicit methods based on strong stability preserving multistep time discretizations

TL;DR

The paper addresses designing robust IMEX methods by leveraging second-order SSP multistep discretizations and performing a linear stability analysis via a scalar test equation. By constructing implicit integrators and analyzing stability through the mapping \varphi_{\lambda}(z), it identifies a 3-step biased scheme that attains A-stability over the full explicit stability domain, while other variants achieve A(\alpha)-stability with modest angular limits. The results show these SSP-based IMEX schemes have stability properties comparable to traditional CNAB and IMEX BDF2 methods, with trade-offs in memory requirements and potential nonlinear stability considerations. The work provides guidance on selecting IMEX schemes for advection-diffusion-type PDEs and suggests avenues for extending analysis to higher-order SSP variants and IMEX Runge-Kutta formulations.

Abstract

In this note we propose and analyze novel implicit-explicit methods based on second order strong stability preserving multistep time discretizations. Several schemes are developed, and a linear stability analysis is performed to study their properties with respect to the implicit and explicit eigenvalues. One of the proposed schemesis found to have very good stability properties, with implicit A-stability for the entire explicit stability domain. The properties of the other proposed schemes are comparable to those of traditional methods found in the literature.

Figures (5)

• Figure 1: Explicit stability domain, $\mathcal{S}$, for the two SSP time discretizations \ref{['eq:explicit-2nd-order-ssp-schemes']}.
• Figure 2: Stability regions, $\mathcal{D}$, for the implicit integrators \ref{['eq:implicit-integrators']}, where $k=3$ and $\beta=0$. The methods are stable outside the shaded regions.
• Figure 3: Image of the unit circle under the mapping $\varphi_{\lambda}(z)$, with $\lambda \in \partial\mathcal{S}$, for the schemes given by \ref{['eq:imex-scheme-biased']} with $k=3$. The implicit stability domain of the method is outside the shaded region.
• Figure 4: Image of the unit circle under the mapping $\varphi_{\lambda}(z)$, with $\lambda \in \partial\mathcal{S}$, for the schemes given by \ref{['eq:imex-scheme-biased']} with $k=4$. The implicit stability domain of the method is outside the shaded region.
• Figure 5: Explicit stability region $\mathcal{S}$ for the time discretization \ref{['eq:explicit-2nd-order-ssp-schemes-k3']}, with the Fourier symbol of the third order $\kappa=1/3$ finite difference advection scheme ($\sigma=0.35$), and the bound ($\gamma=0.5$) that defines the restricted stability region $\mathcal{S}^{\gamma}$ in Lemma \ref{['lemma:m3p2-imex-stability']}.