Lagrangian Coordinates for Moving Boundary Problems

Source abstract

Changes of density which occur naturally in phase change problems introduce movement of bulk material. It is customary in analyzing such problems to ignore this unpleasant complication and consider the densities to be equal. But for one-dimensional problems the complexities introduced by this bulk movement can easily be circumvented. The key idea is posing the problem in local coordinates which are fixed in each phase. We show how to define suitable moving coordinates and, using them, pose and give an explicit solution for a one-dimensional, multi-phase Stefan problem with phases of distinct densities. This explicit solution is essentially a similarity solution in the local coordinates. However, the local coordinates could be used with finite element or finite difference schemes to analyze problems for which similarity solutions do not exist.