© 2022A moving boundary problem is numerically investigated with gallium substance and FVM (finite volume method). Following the determination of the optimal grid size and time-step, a natural convection problem in a vertical slot is studied to validate the multi-cellular melting front. Different heating configurations are modelled to understand the solid-liquid phase change phenomenon under different scenarios for the gallium melting problem. These different scenarios include spatial differential heating (sinusoidal), temporal differential heating (pulsating), inclination angle, and MHD conditions. It is found that grid independence study should not be conducted with liquid fraction or similar average value. If the first time-step is larger than 0.1 s, early development of the Bénard cells will not be captured accurately, even when the outer iterations converge. It is also observed that if peak point of differential heating is close to the top of the domain, melting front will progress faster. When Lorentz force is active, at high Hartmann numbers, oscillation of the Nusselt number is dampened.