An Analysis of the Blended Three-Step Backward Differentiation Formula Time-Stepping Scheme for the Navier-Stokes-Type System Related to Soret Convection

Abstract

In this article, we investigate the stability and convergence of a new class of blended three-step Backward Differentiation Formula (BDF) time-stepping scheme for spatially discretized Navier-Stokes-type system modeling Soret driven convective flows. A Galerkin mixed finite element spatial discretization is assumed, and the temporal discretization is by the implicit blended threestep BDF scheme. The blended BDF scheme is more accurate than the classical second order accurate two-step BDF (BDF2) scheme, yet strongly A-stable. We consider an implicit, linearly extrapolated version of the scheme to improve its efficiency. We present optimal finite element error estimates and prove the scheme is unconditionally stable and convergent. Numerical experiments are presented that compare the scheme to the classical BDF2 scheme.

Long time stability of a linearly extrapolated blended BDF scheme for multiphysics flows

“…Among all the multiphysics flow examples issued in this study, the most challenging kind of flow is the double-diffusive convection case due to the highly oscillatory nature of its solutions [7,16,17]. As the Rayleigh number increases, complex behavior of the solutions becomes more visible.…”

“…In [17], Ravindran considers the stability and convergence of a double diffusive convection system with the blended BDF scheme in short time intervals. The current study attempts to extend the works above to study the notion of long time stability by combining the BLEBDF idea and its effects on several multiphysics flows such as Navier-Stokes, natural convection and double-diffusive convection.…”