Optimal convergence analysis of second‐order time‐discrete stabilized scheme for the thermally coupled incompressible MHD system

Abstract In this paper, we construct an optimal convergence analysis of second‐order stabilized scheme for the thermally coupled incompressible magnetohydrodynamic (MHD) system. First, we construct first‐ and second‐order time‐discrete schemes in which the time derivative term is treated by the first‐order backward Euler method and the second‐order backward difference formulation, respectively. And the nonlinear terms are treated by semi‐implicit method. Importantly, we use the Gauge–Uzawa method to decouple velocity and pressure. The proposed schemes have the following two distinct features: they do not need to give an initial value of the pressure, and they do not require artificial boundary conditions on the pressure. Second, the unconditional energy stability of the two schemes is proved. Then, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes.