A fixed-grid finite volume numerical approach is developed to simulate the melting during the solid-liquid phase-change driven by convection as well as by conduction. This approach adopts the enthalpy-porosity method augmented with the front-layer predictor-corrector and the pseudo Newton-Raphson algorithms that were devised to track the phase front efficiently in the conduction-driven phase-change problems. The computational results compare well with experimental data and transformed-grid results in the literature. Also, the effect of the delayed heat-up at a heated wall on the melting process is investigated.