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The classical Stefan problem is reduced as the singular limit of phase-fieldequations. These equations are for temperature $u$ and the phase-field$\varphi$, consists of a heat equation: $$ u_t+\ell\varphi_t=\Delta u, $$ and aGinzburg-Landau equation: $$ \epsilon\varphi_t=\epsilon\Delta\varphi-\frac{1}{\epsilon}W^\prime (\varphi )+\ell (\varphi )u, $$ where $\ell$ is alatent heat and $W$ is a double-well potential whose wells, of equal depth,correspond to the solid and liquid phases. When $\epsilon\to 0$, the velocityof the moving boundary $v$ in one dimension and that of the radius in thecylinder or sphere are shown as the following Stefan problem,\\ $$ \left\{\begin{array}{l} u_t-\Delta u =0\\\\ \displaystylev=\frac{1}{2}\left[\frac{\partial u}{\partial n}\right]_\Gamma \displaystyleu=-\frac{m}{2\ell}[\kappa -\alpha v]_\Gamma \end{array} \right. $$ where$\alpha$ is a positive parameter, $[\frac{\partial u}{\partial n}]_\Gamma$ isthe jump of the normal derivatives of $u$ (from solid to liquid), and$m=\int_{-1}^1\left(2W(\varphi)\right)^{1/2}d\varphi$. Since it is sufficientto describe the phase transition of single component by the phase-fieldequation, we analyze the phase-field equation, and investigate whether theequation shows the Stefan problem or not. The velocity of the moving boundaryin the cylinder and sphere are determined and the result of the simulation ofthe equation is also presented. Next, we consider the velocity of interfacewhich depends on the temperature. The control of width of diffusion layer bythe parameter of phase-field equations is investigated in order to realize thesingular limit of phase field equations by the numerical method.

Phase field equation in the singular limit of the Stefan problem