An Extrapolated Second Order Backward Difference Time-Stepping Scheme for the Magnetohydrodynamics System

In this article, we introduce a fully implicit, linearly extrapolated second-order backward difference time-stepping scheme for solving a time dependent non-homogeneous magnetohydrodynamic system for electrically conducting fluids. The extrapolated time-stepping scheme is used for time discretization and the mixed finite element method is used for spatial discretization. We first prove unconditional energetic stability without introducing an undesirable exponential Gronwall constant. Complete error analysis is provided without assuming any stability condition or restrictions on the time-step size. Numerical experiments are presented to confirm the theoretical convergence results and efficiency of the scheme.