Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nédélec edge element for the magnetic induction and the equal order Lagrange elements for the thermal equation. This fully discrete scheme only needs to solve a linear system at each time step, and the system is unique solvable. We prove the proposed scheme is unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present optimal error estimates for the velocity, magnetic induction and temperature. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.