A methodology is set forth for the numerical solution of transient two-dimensional freezing of a phase change medium contained within a closed vertical tube. The transient is initiated when the containment tube, which is initially filled with a liquid whose temperature is above the fusion value, is exposed to an external fluid environment having a below-fusion temperature. The solution domain encompasses the liquid and solid phases and the tube wall. Natural convection occurs in the liquid phase. Coordinate transformations are employed to immobilize and to straighten the moving, curved interface which separates the solid and liquid phases. The finite-difference equations for the temperature and velocity fields are derived using a con-trot volume approach and an implicit treatment of the time wise variations. The energy balance at the solid-liquid interface is discretized explicitly to provide a means for determining the movement of the interface as freezing proceeds. Numerical results obtained by applying the methodology are presented in a companion paper which follows.