Stability analysis of consistent splitting implicit-explicit multistep methods up to ninth-order accuracy for incompressible flows

Abstract

This work presents a concise, unified stability theory of high-order decoupled \lan{implicit-explicit linear multistep (IELM)} methods based on the well-known consistent splitting technique for the incompressible Navier-Stokes equation. With the help of the recent semi-generating function approach and the global discrete energy analysis, one can establish the unconditional stability of a consistent splitting IELM method with respect to the $\ell^{\infty}(H^1)\cap \ell^{2}(H^2)$ norm if the associated implicit-explicit controllability intensity is larger than $\sqrt{2}/2$, a constant determined by the Stokes pressure estimate. It is shown that the $β$-parameterized GBDF-$\rmk$ ($2\le \rmk\le5$) schemes and $γ$-parameterized SIELM-$\rmk$ ($2\le \rmk\le9$) schemes can fulfill this requirement of implicit-explicit controllability intensity by choosing proper parameters so that they can theoretically maintain the unconditional stability of the associated consistent splitting IELM methods. Numerical experiments are also included to support our theory.