Efficient ZEC-DG numerical framework for thermally-coupled MHD system and simulations of magnetically controlled Rayleigh-Bénard convection

In this paper, we consider numerical approximations for a modified thermally-coupled incompressible magneto-hydrodynamic (MHD) system, which integrates the governing equations of MHD flow with a heat transport equation. By using the techniques of the modified pressure and auxiliary variables, we first reformulate the model into a form that is suitable for numerical approximation and establish its energy dissipation law. Then, we develop a novel unconditionally energy-stable, fully decoupled, linear, second-order time-accurate scheme for this highly coupled and nonlinear system. The proposed scheme is constructed by combining the ZEC (zero-energy-contribution) method, the DG (discontinuous Galerkin) method, and a second-order Gauge-Uzawa projection method. A rigorous energy stability analysis confirms the unconditional stability of the proposed scheme, along with a detailed description of its numerical implementation. To show the robustness of the proposed method, a series of comprehensive numerical simulations are carried out, including 2D and 3D Rayleigh-Bénard convection, as well as 3D natural convection driven by sidewall heating. The numerical results demonstrate the scheme’s ability to capture key physical phenomena, including thermal plume evolution, symmetry breaking induced by anisotropic magnetic fields, and the direction-dependent modulation of convective intensity, namely, enhancement under horizontal magnetic forcing and suppression under vertical magnetic forcing.