A thermodynamically consistent numerical method for a phase field model of solidification

• A numerical method which unconditionally preserves the two laws of thermodynamics.
• The ideas can be extended to other phase field problems.
• Proof of the Lyapunov stability in the discrete and continuous models.
• Simulations show that the method is less diffusive than the midpoint rule.

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.