Abstract The simulation of melting phenomena in the frame work of continuum theory can be handled through many different approaches, among which the fixed-grid methods are the most common ones. Even though such methods have been in use for several decades, only scarce publications exist addressing their performance when solving moving boundary problems. Recently, five of the most used energy formulations with a strong coupling between temperature and enthalpy have been studied in an extensive benchmark both qualitatively and quantitatively. The present paper extends this work by including the role of the formulation of the convective term in the energy equation. Two cases are considered: a first one where all the enthalpy is convected, while in the second one it is the sensible enthalpy only. Similarly to what was done in the aforementioned study, this analysis also includes investigations concerning the modeling of the phase transition, through the associated temperature range for the mushy zone, and several numerical parameters, namely the time step, the mesh coarsening, the CFL condition and the tolerance for the energy equation. The numerical results are compared to a well known quasi-2D melting experiment from the literature in order to determine clearly wrong results. In addition, the liquid fraction and its variance are analyzed for every solver. Except for the solver using an apparent heat capacity method, all of them give reasonable results for a broad range of parameters. The sensible enthalpy convection formulation is in general more stable than the all enthalpy convection formulation. For the chosen parameters and the most stable solver - which is practically not affected by the tolerance, the CFL or the maximum time step - it appears that parameters having the strongest influence on the liquid fraction are the width of the mushy zone followed by the mesh and the convective formulation.