Unconditional Long Time $\text{H}^1$-Stability of a Velocity-Vorticity-Temperature Scheme for the $2\text{D}$-Boussinesq System

Abstract

This paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time $\text{H}^1$-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.