An important class of melting problems is that in which the heat required to melt the material is supplied by a distribution of heat sources throughout the volume of the material (e.g. Joule heating) rather than at the boundaries. In such problems the melting does not necessarily take place across a surface; indeed the application of the heat may give rise to a “mushy region” i.e. a volume of the material which is at the melting temperature and in which the solid and liquid phases co-exist. As an aid to the discussion of such problems the weak solution of a heat conduction problem with a phase change is described. Motivated by this definition a finite difference scheme for the phase change problem is then described. It is shown that the solution of this finite difference scheme converges to the weak solution of the heat conduction problem. Finally, the method is applied to the problem of resistance spot welding of thin steel sheets.