Recent changes in weather patterns have made modeling of freezing and thawing of groundwater in cold regions of particular interest to study and model. The Stefan problem models ice and water domains and the free boundary between these domains. The interface condition on the free boundary involves a jump of the heat flux; this is modeled by Heaviside function in the nonlinear evolution equation written for the entire water and ice domain. Replacing the Heaviside function in the Stefan problem with an order parameter leads to a coupled equation called a phase-field model. The phase-field model is of interest on its own since it is a gradient flow of a convex functional augmented by a nonconvex destabilizing term.
In this talk, numerical approaches to solving this phase-field and Stefan problem system are discussed. This includes discussion of numerical schemes for the phase field model including various schemes which attempt to compensate for the lack of convexity and stabilize the scheme without having to resolve a fully implicit problem. We also discuss the nonsmooth phase field model and how to solve the parabolic variational inequality. Current work towards applying this model to permafrost will also be discussed.