Abstract :  We study numerical methods for  the time-dependent magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation, in which the Joule effect and Viscous heating are taken into account.  To overcome the difficulties of  very low regularity of  the heat source terms, a regularized weak system is proposed to deal with Joule and Viscous heating terms.  We  consider an Euler semi-implicit  semi-discrete scheme  for the regularized system.  As both discrete parameter and regularization parameter tend to zero, we prove that the discrete solution converges to a weak solution of the original problem. Next, we consider the fully discrete Euler semi-implicit scheme based on  the mixed finite method to approximate the  fluid  equation and N$/mathrm{/acute{e}}$d$/mathrm{/acute{e}}$lec edge element to  the magnetic induction. The fully discrete scheme requires only solving a linear system per time step.   The error estimates for  the velocity, magnetic induction and temperature are derived under  a proper regularity  assumption for the exact solution. Finally, several numerical examples are performed to demonstrate  both accuracy and efficiency of our proposed scheme.