Prof.Cheng Wang ,University of Massachusetts Dartmouth, United States

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A temporally second-order accurate, fully discrete finite element method is proposed and analyzed for the magnetohydrodynamic (MHD) equations. A modified Crank-Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection
term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, a decoupling projection method of the Van Kan type is used in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled
Stokes solver needs to be analyzed in details. Optimal-order convergence is proved for the proposed decoupled projection finite element scheme. Numerical examples are provided to illustrate the theoretical results.