Two-phase Stefan problem with smoothed enthalpy

J. Shen (Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, and School of Mathematical Sciences, Xiamen University, Xiamen, China)

Abstract

The enthalpy regularization is a preliminary step in many numerical methods for the simulation of phase change problems. It consists in smoothing the discontinuity (on the enthalpy) caused by the latent heat of fusion and yields a thickening of the free boundary. The phase change occurs in a curved strip, i.e. the mushy zone, where solid and liquid phases are present simultaneously. The width $\epsilon$ of this (mushy) region is most often considered as the parameter to control the regularization effect. The purpose we have in mind is a rigorous study of the effect of the process of enthalpy smoothing. The melting Stefan problem we consider is set in a semi-infinite slab, heated at the extreme-point. After proving the existence of an auto-similar temperature, solution of the regularized problem, we focus on the convergence issue as $\epsilon \to 0$. Estimates found in the literature predict an accuracy like $\sqrt{\epsilon}$. We show that the thermal energy trapped in the mushy zone decays exactly like $\sqrt{\epsilon}$, which indicates that the global convergence rate of $\sqrt{\epsilon}$ cannot be improved. However, outside the mushy region, we derive a bound for the gap between the smoothed and exact temperature fields that decreases like $\epsilon$. We also present some numerical computations to validate our results.

Keywords

Stefan problem, phase change problems, enthalpy, convergence

2010 Mathematics Subject Classification

35B06, 65L20, 65N12, 80A20, 80A22

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Published 12 August 2016