Energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models

Abstract

We present a numerical scheme that preserves the total energy and the entropy production rate, termed the energy and entropy production rate preserving scheme, for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics. The scheme is second order in time, linear and energy and entropy production rate preserving for any time steps. The scheme is first discretized in time aided by the energy quadratization (EQ) method and then in space using compact finite difference methods. The linear system resulting from the scheme is shown to be uniquely solvable at both the semi-discrete and the fully discrete level. Mesh refinement tests are performed to show the second-order time convergence rate in the scheme. Several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme. The effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver. The approach to developing the energy and entropy production rate-preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models.