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The aim of this paper is to develop an unconditionally stable Euler implicit/explicit scheme for the natural convection equations based on an introduced exponential scalar auxiliary variable. Our numerical scheme involves treating the linear terms in the implicit scheme and nonlinear terms in the explicit scheme. Then, the considered problem is decoupled into a series of linearized subproblems, and these algebraic systems with constant coefficient matrixes can be solved efficiently. Compared with the existing theoretical findings of the implicit/explicit scheme for the incompressible flow [He YN. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or nonsmooth initial data. Math Comput. 2008;77(264):2097–2124.; He YN, Huang PZ, Feng XL. H2-stability of first order fully discrete schemes for the time-dependent Navier-Stokes equations. J Sci Comput. 2015;62:230–264; He YN, Li J. A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math. 2010;235(3):708–725; Jin JJ, Zhang T, Li J. H2 stability of the first order Galerkin method for the Boussinesq equations with smooth and nonsmooth initial data. Comput Math Appl. 2018;75(1):248–288; Su J, He YN. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete Continuous Dyn Syst Ser B. 2017;22(9):3421–3438], our analysis removes the restriction on time step, and obtains the unconditional stability and convergence theoretical findings. Some numerical results are also provided to verify the established theoretical analysis and show the performances of the considered numerical scheme.

Fully decoupled, linear and unconditional stability implicit/explicit scheme for the natural convection problem