Numerical analysis and computation of a type of IMEX method for the time-dependent natural convection problem

A new numerical regularization method for the natural convection problem is presented, which is based on a type of implicit-explicit (IMEX) second-order time-stepping schemes in temporal discretization and stabilized mixed finite element in spatial discretization. This method deals with a non-linear advection term in both the momentum equation and the energy equation by linearization. We only need to solve a linear problem at each time step and the discrete curvature of the solutions is added as a stabilization term for the velocity, the pressure and the temperature, respectively. Unconditional stability is proved and an a priori error estimate is derived. Finally, a series of numerical experiments are also given to confirm the theoretical analysis and to demonstrate the efficiency of the new method.