In this work, we consider two of the most frequently used two-dimensional nonlinear time-fractional anomalous sub-diffusion models for simulating transport phenomena in heterogeneous binary media—a variable-order model and a generalised transport equation based on the Riemann-Liouville fractional operator. Our computational modelling framework uses a second order modified weighted shifted Grünwald-Letnikov scheme with correction terms to approximate the time-fractional derivative and nonlinear source term, together with an unstructured mesh control volume method for the spatial discretisation that accommodates the heterogeneous model properties. The resulting nonlinear system of algebraic equations is then stepped in time using an efficient Jacobian-free, Newton-Krylov solver to determine the set of discrete solution unknowns. We derive a semi-analytical solution and mass balance equation for a class of two-layered problems to evaluate the accuracy of the different computational models. A key contribution of our work is to identify the correct form of the interfacial boundary conditions to impose for the flux terms within the time-fractional framework and to illustrate the significant impact that the time-fractional indices have on the mass transfer. We show that the generalised transport model not only exhibits the correct physical solution behaviour, it produces a more accurate overall mass balance in comparison to the variable-order time-fractional model which is unable to resolve the abrupt changes in the solution behaviour at the interface between two different media having contrasting diffusivity, fractional index and source term. This finding indicates that this generalised model is an effective tool for characterising anomalous transport processes in heterogeneous systems. Finally, a series of numerical examples are presented to verify the theoretical analysis and to highlight the capability of the generalised transport model.

在这项工作中，我们考虑了两个最常用的二维非线性时间分数异常子扩散模型，用于模拟异质二元介质中的传输现象-变阶模型和基于Riemann-Liouville分数的广义传输方程操作员。我们的计算建模框架使用带有修正项的二阶修正加权移位Grünwald-Letnikov方案来近似时间分数导数和非线性源项，以及用于空间离散化的非结构化网格控制体方法，该方法可以适应异构模型的特性。然后，使用高效的无Jacobian，Newton-Krylov求解器及时逐步生成所得的代数方程组非线性系统，以确定离散的未知量集合。我们推导了一类两层问题的半解析解和质量平衡方程，以评估不同计算模型的准确性。我们工作的关键贡献在于确定界面边界条件的正确形式，以便在时间分数框架内施加通量项，并说明时间分数指数对质量传递的重大影响。我们表明，广义输运模型不仅表现出正确的物理溶液行为，而且与无法解决界面处溶液行为的突然变化的变阶时间分数模型相比，它还能产生更准确的整体质量平衡。在两个不同的介质之间具有相对的扩散率，分数指数和源项。这一发现表明，该通用模型是表征异构系统中异常传输过程的有效工具。最后，通过一系列数值例子验证了理论分析并突出了广义运输模型的能力。