Decoupled and Unconditionally Energy Stable Numerical Schemes for the Thermally Coupled Incompressible Magnetohydrodynamics Flow (Advances in Applied Mathematics and Mechanics, 2025)
This paper proposes two efficient numerical schemes for the time dependent thermally coupled incompressible Magnetohydrodynamics (TMHD) equations. Firstly, the existence and uniqueness of the weak solutions are established by employing the Galerkin method and extracting the subsequences. Secondly, the Euler semi-implicit scheme is designed for the target problem. The energy dissipation and stability of the numerical scheme are developed, and the optimal error estimates are also presented. Thirdly, the implicit/explicit (IMEX) scheme is utilized to simplify the computational complexity, then the aimed problem splits into three linear elliptic subproblems with the constant coefficients at each time level $t_n$. Meanwhile, the "zero-energy contribution" (ZEC) approach is used to surmount the restriction $Δ t≤ C$ caused by the IMEX scheme. The corresponding mathematical findings, including the energy dissipation and stability of the IMEX-ZEC scheme and the convergence of the numerical solutions are obtained via the energy method and the Gronwall lemma. Finally, we conduct some numerical results to confirm the established theoretical findings and show the performances of the considered numerical schemes.