Fully decoupled, linear and unconditionally energy stable time discretization scheme for solving the magneto-hydrodynamic equations

Abstract

In this paper, we consider numerical approximations for solving the magneto-hydrodynamic equations, which couples the Navier–Stokes equations and Maxwell equations together. A challenging issue to solve this model numerically is the time discretization, 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 linear, fully decoupled first order time marching scheme, by combining the projection method for Navier–Stokes equations 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 of the semi-discretization rigorously. Various numerical simulations are implemented to demonstrate the stability and the accuracy.