Fully Decoupled, Second-Order Accurate and Unconditionally Energy Stable Numerical Scheme for the Boussinesq Equations

Abstract:

      In this paper, we construct a fully decoupled, second-order semi-discrete numerical scheme for the Boussinesq equations based on the scalar auxiliary variable (SAV) approach. Firstly, the original Boussinesq system is transformed into an equivalent Boussinesq system by introducing scalar auxiliary variables. Secondly, a time marching scheme based on the second-order backward differentiation formula (BDF2) and the pressure-correction method is developed, where the velocity and pressure are decoupled. Thirdly, we use the scalar auxiliary variable to decompose each discrete equation into several constant-coefficient sub-equations according to the splitting technique. Hence, one only needs to solve few decoupled constant-coefficient elliptic equations at each time step. We rigorously prove unconditional energy stability and unique solvability of the discrete scheme. Furthermore, we provide a detailed implementation of the decoupling procedure. Finally, various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed scheme.