# Numerics for the two-phase Stefan problem

## Norbert Pozar

## Introduction

The following videos show the evolution of enthalpy \(h\) in the two-phase Stefan problem with no-flux boundary in 2D: \[ \begin{cases} h_t(x,t) - \Delta \chi(h(x,t)) = f(x,t), & \text{in } \Omega \times (0, T)\\ \frac{\partial \chi(u)}{\nu} = 0, & \text{on } \partial \Omega \times (0,T)\\ h(x,0) = h_0(x), \end{cases} \] where \(h\) is the enthalpy, \(f\) is the volumetric heat source and \(\chi(h)\) is the temperature. The function \(\chi\) is defined as \[ \chi(s) = \min(s, \max(0, s -1)) = \begin{cases} s & s < 0,\\ 0 & 0 \leq s \leq 1 & s-1 & s > 1, \end{cases} \] see figure 1.

The solutions contain the following regions:

*solid*– red*liquid*– yellow*mushy region*– blue

Images were produced using Mathematica 7 and encoded using mencoder. To view these videos, you need to be able to play videos from youtube.