Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

中文翻译：

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多维互扩散：数学模型、数值模拟和实验

在过去的二十年里，扩散计算取得了巨大进步，如今材料的许多关键特性都可以通过建模和模拟来准确预测。在本文中，我们首次从模型、模拟和实验三个维度对相互扩散进行了综合研究。该模型遵循 Vegard 规则的局部质量守恒，并与 Darken 的双速度方法相结合。该方法使用具有初始和非线性耦合边界条件的强耦合微分方程的非线性抛物线-椭圆系统表示。保留 Vegard 规则的隐式有限差分方法是通过一维和二维情况下的一些线性化和分裂思想生成的。研究了隐差分格式解的存在唯一性和差分方法的一致性定理。将数值结果与三元 Fe-Co-Ni 系统的实验数据进行比较。揭示了两组的良好一致性，这证实了该方法的强度。