On the Long-Time $H^1$-Stability of the Linearly Extrapolated BDF2 Time-Stepping Scheme for Coupled Multiphysics Flow Problems (Communications in Mathematical Research, 2025)
The purpose of the current article is to study the $H^1$-stability for allpositive time of the linearly extrapolated BDF2 time-stepping scheme for themagnetohydrodynamics and Boussinesq equations. Specifically, we discretizein time using the linearly backward differentiation formula, and by employingboth the discrete Gronwall lemma and the discrete uniform Gronwall lemma,we establish that each numerical scheme is uniformly bounded in the $H^1$-norm.
Keywords: Magnetohydrodynamics equations; Boussinesq equations; linearly extrapolated BDF2 timestepping scheme; long-time stability.