The problem of a spatially discontinuous diffusion coefficient ($D(\boldsymbol x)$) is one that may be encountered in hydrogeologic systems due to natural geological features or as a consequence of numerical discretization of flow properties. To date, mass-transfer particle-tracking (MTPT) methods, a family of Lagrangian methods in which diffusion is jointly simulated by random walk and diffusive mass transfers, have been unable to solve this problem. This manuscript presents a new mass-transfer (MT) algorithm that enables MTPT methods to accurately solve the problem of discontinuous $D(\boldsymbol x)$. To achieve this, we derive a semi-analytical solution to the discontinuous $D(\boldsymbol x)$ problem by employing a predictor-corrector approach, and we use this semi-analytical solution as the weighting function in a reformulated MT algorithm. This semi-analytical solution is generalized for cases with multiple 1D interfaces as well as for 2D cases, including a $2 \times 2$ tiling of 4 subdomains that corresponds to a numerically-generated diffusion field. The solutions generated by this new mass-transfer algorithm closely agree with an analytical 1D solution or, in more complicated cases, trusted numerical results, demonstrating the success of our proposed approach.

中文翻译：

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一种模拟不连续扩散系数输运的传质粒子跟踪方法

空间不连续扩散系数 ($D(\boldsymbol x)$) 的问题是水文地质系统中由于自然地质特征或流动特性数值离散化可能遇到的问题。迄今为止，传质粒子跟踪 (MTPT) 方法是一系列拉格朗日方法，其中通过随机游走和扩散传质联合模拟扩散，一直无法解决这个问题。这份手稿提出了一种新的传质 (MT) 算法，该算法使 MTPT 方法能够准确解决不连续的 $D(\boldsymbol x)$ 问题。为了实现这一点，我们通过采用预测器-校正器方法推导出不连续 $D(\boldsymbol x)$ 问题的半解析解，并且我们使用该半解析解作为重新制定的 MT 算法中的加权函数。这种半解析解决方案适用于具有多个 1D 界面的情况以及 2D 情况，包括对应于数值生成的扩散场的 4 个子域的 $2 \times 2$ 平铺。这种新的传质算法生成的解与一维解析解或在更复杂的情况下与可信数值结果非常吻合，证明了我们提出的方法的成功。