A streamline diffusion method for nonstationary incompressible magnetohydrodynamics system with variable density

This paper proposes a streamline diffusion numerical method for the unsteady magnetohydrodynamic flows with variable density, in which the electric conductivity and viscosity coefficients depend on the density. We employ stable mini-elements to discretize the Navier-Stokes equations and the Nédélec edge elements to discretize the magnetic induction. In each time step, the equations for velocity and pressure are decoupled with pressure computed by solving one Poisson equation. The streamline diffusion method is applied to the continuity equation, based on which a stable finite element scheme is proposed. We show that the fully discrete numerical scheme is energy-stable and well-posed. Rigorous error estimate of all the variables is established without assuming that the density has a good approximation, which shows that this method performs as well as the constant density.