  We study the evolution of a melting front between the solid and liquid phases
of a pure incompressible material where fluid motions are driven by unstable
temperature gradients. In a plane layer geometry, this can be seen as classical
Rayleigh-B\'enard convection where the upper solid boundary is allowed to melt
due to the heat flux brought by the fluid underneath. This free-boundary
problem is studied numerically in two dimensions using a phase-field approach,
classically used to study the melting and solidification of alloys, which we
dynamically couple with the Navier-Stokes equations in the Boussinesq
approximation. The advantage of this approach is that it requires only moderate
modifications of classical numerical methods. We focus on the case where the
solid is initially nearly isothermal, so that the evolution of the topography
is related to the inhomogeneous heat flux from thermal convection, and does not
depend on the conduction problem in the solid. From a very thin stable layer of
fluid, convection cells appears as the depth -- and therefore the effective
Rayleigh number of the layer increases. The continuous melting of the solid
leads to dynamical transitions between different convection cell sizes and
topography amplitudes. The Nusselt number can be larger than its value for a
planar upper boundary, due to the feedback of the topography on the flow, which
can stabilize large-scale laminar convection cells.
