We consider numerical approximations for solving the nonlinear magneto-hydrodynamic equations, which couples the Navier-Stokes equations and Maxwell equations together. A challenging issue to solve this model numerically is about the time marching problem, i.e., how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level. We solve this issue in this paper by developing a fully decoupled, first order time stepping scheme, by combining the projection method and some subtle implicit-explicit treatments for nonlinear coupling terms. We further prove that the scheme is unconditional energy stable and derive the optimal error estimates rigorously. Various numerical experiments are implemented to demonstrate the stability and the accuracy in simulating some benchmark simulations, including the Kelvin-Helmholtz shear instability and the magnetic-frozen phenomenon in the lid-driven cavity.

Dr. Yang Xiaofeng received his Ph. D in Mathematics, Purdue University in 2007, and his B.S. and M.S. in Mathematics from University of Science and Technology of China. He was a Postdoctoral Research Associate in the Department of Mathematics, University of North Carolina at Chapel Hill (UNC-CH), for the year 2007-2009 and joined Department of Mathematics at University of South Carolina in fall 2009 as a Tenure-Track Assistant Professor. He has been promoted to full professor in Dec. 2017. His research areas are Applied and Computational Mathematics, Mathematical modeling and scientific computing with applications to Soft Matter/Complex Fluids/Cell Dynamics, and Numerical analysis of finite element methods and spectral methods.