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SUMMARYA two-dimensional incompressible magneto-hydrodynamic code is presented in order to solve the steady state or transient magnetized or neutral convection problems with the effect of heat transfer. The code utilizes a numerical matrix distribution scheme that runs on structured or unstructured triangular meshes and employs a dual time-stepping technique with multi-stage Runge-Kutta algorithm. The code can be used to simulate the natural convection with internal heat generation and absorption and nonlinear timedependent evolution of heated and magnetized liquid metals exposed to external fields. MAGNETO-HYDRODYNAMIC EQUATIONSIn this work, it is assumed that the flow is characterized by two-dimensional (2D) incompressible Navier-Stokes-Fourier plus Maxwell equations in the quasi-static approximation (i.e. Magnetohydrodynamic (MHD) equations). The MHD system is coupled with the temperature effects through gravitational force by means of the Bousinessq approximation. In addition, the fluid is assumed to be electrically conducting and locally quasi-neutral at the same time. As a consequence, not only the fluid flow contributes to the electric and magnetic field distributions in the medium but

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