Numerical analysis of an improved projection method for the evolutionary magnetohydrodynamic equations with modular grad-div stabilization

In this paper, an improved projection scheme has been constructed for solving the nonstationary magnetohydrodynamic equations, based on the modular grad-div stabilization. Owing to the utilization of the projection scheme and some delicate implicit-explicit approach to the nonlinear coupled terms, the proposed scheme is linear, decoupled and unconditionally energy stable. In addition, the modular grad-div stabilization technique is introduced to improve the conservation of Gauss's law and mass, and reduce the splitting error of the projection method, which is simple to implement with a small intrusion step into existing code. In this way, the developed scheme can prevent the solver from breakdown and enhance computational efficiency with the increasing of grad-div parameters. What's more, we present a rigorous convergence analysis. Finally, several numerical experiments are performed to illustrate the theoretical consequences and their effectiveness.