Enthalpy method for phase change

Definition

Let $\Omega \subseteq \R^n$ be a domain filled with a material that melts at temperature $T_m$. Assume for simplicity that solid and liquid have the same density $\rho > 0$, specific heat $c > 0$ and thermal conductivity $k > 0$. Let $L > 0$ be the latent heat per unit mass. Let $H$ be the enthalpy per unit volume, measured from the solid at $T_m$. The temperature as a function of enthalpy is $$\beta(H) = \begin{cases} T_m + H / (\rho c), & H < 0, \\ T_m, & 0 \le H \le \rho L, \\ T_m + (H - \rho L)/(\rho c), & H > \rho L. \end{cases}.$$ The enthalpy formulation of the Stefan problem is the equation $$\frac{\partial H}{\partial t} = \nabla \cdot \big( k \nabla \beta(H) \big) \text{ in } \Omega \times (0, \infty).$$ It is understood in the weak sense, with the initial enthalpy and boundary conditions for $T = \beta(H)$. The enthalpy method is the numerical solution of this single equation for $H$ on a fixed grid, with the temperature recovered as $T = \beta(H)$. The solid region is $\set{H < 0}$, the liquid region is $\set{H > \rho L}$, and the set $\set{0 \le H \le \rho L}$ contains the phase boundary.