A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations

For highly coupled nonlinear incompressible magnetohydrodynamic (MHD) system, a well-known numerical challenge is how to establish an unconditionally energy stable linearized numerical scheme which also has a fully decoupled structure and second-order time accuracy. This paper simultaneously reaches all of these requirements for the first time by developing an effective numerical scheme, which combines a novel decoupling technique based on the “zero-energy-contribution” feature satisfied by the coupled nonlinear terms, the second-order projection method for dealing with the fluid momentum equations, and a finite element method for spatial discretization. The implementation of the scheme is very efficient, because only a few independent linear elliptic equations with constant coefficients need to be solved by the finite element method at each time step. The unconditional energy stability and well-posedness of the scheme are proved. Various 2D and 3D numerical simulations are carried out to illustrate the developed scheme, including convergence/stability tests and some benchmark MHD problems, such as the hydromagnetic Kelvin-Helmholtz instability, and driven cavity problems. •A first decoupling fully-discrete scheme for the MHD model is developed.•The scheme is second-order time accurate, linear and unconditionally energy stable.•A few independent elliptic constant-coefficient equations are needed to be solved.•The energy stability and well-posedness of the scheme are strictly proved.•A series of numerical examples of accuracy/stability, benchmark simulations are given.