AbstractWe present numerical solutions for the following Stefan problems. The half‐spacez>0 is initially filled with liquid at its fusion temperature. The boundaryz= 0, taken to be thexaxis, is maintained at a constant temperature, less than the fusion temperature, forx0, the first problem considers the case of an insulated boundary, and the second problem considers the case of the boundary maintained at the fusion temperature. This gives rise to a solid‐liquid interface curved in the (x,z) pl