Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a two-dimensional grid of rectangular blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.

翻译：由于解决微规模异质的计算成本,通过多种介质直接进行数字模拟扩散可能比较困难。克服这一困难的方法之一是将模型同化,用从适当边界值问题的解决方案中计算出的有效差异性来取代空间变化的微规模细软体。在本文中,我们提出了一个新的半分析方法,以解决这一边界值问题,并计算混杂介质的有效差异性。我们将我们的新解决方案方法与标准限量体积方法进行比较,并表明在几个标准测试案例的计算时间较少的情况下可以实现等值精度。我们还演示了如何将新解决方案方法应用于由两维矩形区块格代表的复杂多变性。这些结果表明,我们新的半分析方法有可能大大加快在多种介质中进行扩散的模拟。