The stability and convergence of the Euler implicit/explicit scheme for the time-dependent thermally coupled incompressible MHD problem

Abstract

The time-dependent thermally coupled incompressible magnetohydrodynamic (TMHD) equations consists of the incompressible \linebreak MHD problem with a nonlinear parabolic equation, and much numerical challenges are caused by the nonlinear terms and the coupling of variables. While the implicit/explicit (IMEX) scheme has several features, such as (1) Transforming the nonlinear problem into the linear problem; (2) Simplifying the computations due to the coefficient matrix of the discrete algebraic equations is constant; (3) Splitting the multi-variables coupling problem into the single variable subproblems; (4) Keeping good stability and accuracy of the numerical approximations. Here, we consider the IMEX scheme for the incompressible TMHD equations based on the finite element method. Firstly, the stability and convergence results of spatial discrete numerical solutions are provided by the energy method and the triangular inequality. Secondly, the stability and optimal error estimates of the numerical approximations in the IMEX numerical scheme are presented by the mathematical induction method and the energy method and the Gronwall lemma. Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical scheme.