there is a very large literature, you could start for example from a [textbook](https://books.google.nl/books/about/The_Stefan_Problem.html?id=l0-t-sjVCqYC&redir_esc=y); an overview of numerical methods is given [here](http://www.sciencedirect.com/science/article/pii/S0377042705003730):

> In this paper, we present a critical comparison of the suitability of
> several numerical methods, level set, moving grid and phase field
> model, to address two well-known Stefan problems in phase
> transformation studies: melting of a pure phase and diffusional
> solid-state phase transformations in a binary system. Similarity
> solutions are applied to verify the numerical results.