Unconditionally energy stable, splitting schemes for magnetohydrodynamic equations

In this article, we propose first-order and second-order linear, unconditionally energy stable, splitting schemes for solving the magnetohydrodynamics (MHD) system. These schemes are based on the projection method for Navier-Stokes equations and implicit-explicit treatments for nonlinear coupling terms. We transform a double saddle points problem into a set of elliptic type problems to solve the MHD system. Our schemes are efficient, easy to implement, and stable. We further prove that time semidiscrete schemes and fully discrete schemes are unconditionally energy stable. Various numerical experiments, including Hartmann flow and lid-driven cavity problems, are implemented to demonstrate the stability and the accuracy of our schemes.