We describe a series of numerical simulations of dissolution-driven convection in a reactive porous medium heated from above. The physical system consists of a porous medium made of the frozen component of a binary mixture that is immersed in a liquid mixture with which it is in thermodynamic equilibrium. Surface heating results in melting of the uppermost material which releases dense solute and drives compositional convection. An interface develops between the upper region, in which the solid matrix has completely melted, and a lower region, in which the frozen solute evolves. The interface descends as melting proceeds. During the numerical simulations, scaled to be similar to previous experiments using potassium nitrate crystals and their saturated aqueous solution (Hallworth, Huppert & Woods, J. Fluid Mech. vol. 535, 2004, p. 255), there are three distinct phases: a purely conductive phase; followed by a phase with very brief, intense, compositionally driven convection; followed by a prolonged phase of more sedate compositionally driven convection in which the average kinetic energy is roughly one order of magnitude less than during the intense early phase. The field equations and the numerical methodology are presented in addition to a simple analytical model for the rate of motion of the interface. The analytical model, valid in the limit of very rapid mixing of the solute, is shown to be in good agreement with the numerical results of purely conductive calculations with a large diffusion coefficient. We investigate solutions for various values of the Rayleigh number and quantify the degree of interface motion as a function of this parameter. These simulations may be particularly applicable to problems associated with post-cumulate processes in magma chambers.