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SUMMARYA block Gauss-Seidel iteration procedure is employed to couple two independent heat transfer codes representing different scales and phenomena in a melt-crystal growth system via shared domain boundaries. In general, such strategies are attractive because of their simplicity for solving problems that may be represented by linking, via shared boundary conditions, existing codes using a modular design. However, these approaches often meet with limited success due to convergence difficulties associated with the overall model. The mathematical framework of fixed-point iterations is employed to assess convergence behaviour for steady-state, non-linear heat transfer models. Analytical forms describe the convergence of one-dimensional problems and provide insight to the behaviour of more complicated, two-dimensional models. Certain implementations that are physically reasonable result in algorithms that will never converge, demonstrating that notions based on physical intuition may not be useful for predicting algorithm performance. A mixing of flux and temperature information along the shared model boundary leads to successful iterative strategies under many conditions.

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We present an overview of mathematical models and their large-scale numerical solution for simulating different phenomena and scales in melt and solution crystal growth. Samples of both classical analyses and state-of-the-art computations are presented. It is argued that the fundamental multi-scale nature of crystal growth precludes any one approach for modeling, rather successful crystal growth modeling relies on an artful blend of rigor and practicality.