Second order, fully decoupled, linear, exactly divergence-free and unconditionally stable discrete scheme for incompressible MHD equations

This article designs a fully decoupled finite element algorithm with second order time-accuracy for the incompressible vector potential magnetohydrodynamic (MHD) system. The novel feature lies in the fact that it naturally produces an exactly divergence-free discretized solution of magnetic induction. The designed algorithm exhibits second-order accuracy, unconditional stability, linearity and fully decoupling. It is implemented by introducing the scalar auxiliary variable (SAV) techniques, combining second-order pressure-correction method, explicit treatment for the nonlinear/coupled terms, and a finite element method for spatial discretization. The effectiveness of this developed algorithm is demonstrated through various three-dimensional numerical simulations, including convergence tests and benchmark issues such as the driven cavity flow, the hydromagnetic Kelvin-Helmholtz instability and the island coalescence problem.