Add to ORKGAbstract: A Chebyshev Tau numerical algorithm is presented to solve the per-turbation equations that result from the linear stability analysis of the convective motion of a fluid layer that appears when an unconfined solid melts in the presence of gravity. The system of equations that describe the phenomenon constitute an eigenvalue problem whose accurate solution requires a robust method. We solve the equations with our method and briefly describe examples of the results. In the limit where the liquid-solid interface recedes at zero velocity the Rayleigh-Bénard solution is recovered. We show that the critical Rayleigh number Rac and the crit-ical wave number ac are monotonically decreasing functions of the rate of melting of the solid. We ...