The Euler implicit/explicit scheme for the Boussinesq equations

In this article, we consider the stability and convergence of the first-order implicit/explicit scheme for the Boussinesq equations. The finite element spatial discretization is based on a MINI element for the velocity and pressure, which satisfies the discrete inf-sup condition, and a linear polynomial for the temperature. The temporal terms are treated by the Euler implicit/explicit scheme, which is implicit for the linear terms and explicit for the nonlinear terms. The advantage of using the implicit/explicit scheme is that a linear system with constant coefficient matrix is obtained, which can save a lot of computational cost. The main novelties of this work are the stability of numerical solutions under the conditions k(1) Delta t <= 1 and k(2) Delta t <= 1 with two positive constants k(1), k(2) and the optimal error estimates of numerical solutions in different norms. Finally, some numerical results are provided to verify the performances of the Euler implicit/explicit scheme.