This study describes a fast-accurate computational procedure for the thermal design of latent heat storage systems with metal foams based on the prediction of the unsteady fluid mechanics, heat transfer and solid-liquid phase change. The analysis considers three PCMs: octadecane, paraffin wax and sodium nitrate, embedded in either aluminum or copper metal foams inside a cubical cavity. The physical-mathematical model used two energy equations, an apparent specific heat-variable viscosity phase change model with a temperature dependent liquid fraction and the Darcy-Brinkman-Forchheimer (DBF) porous model for the convective fluid and heat flows in the metal foam. The finite volume method allowed solving the thermal conjugate model for convective melting in the PCM/metal foams using an improved SIMPLER prediction-correction algorithm. The findings indicate that DBF is the suitable porous model to describe latent heat storage of PCMs in metal foams. Additionally, two energy equations must be used in PCM/metal foams when the interstitial Nusselt number (Nui), calculated in terms of the volumetric heat transfer coefficient, is smaller than 100. The findings highlighted that the numerical prediction of latent heat storage by PCMs in metal foams must be obtained using a third-order accurate calculation of the transient terms in the momentum and energy equations. The results indicated that the high rate of convergence of the innovative SIMPLERnP algorithm allowed a reduction of four times in the time of computation with respect to one of the most reliable and fast prediction-correction schemes for the finite volume method.