We start with a general governing equation for diffusion transport, written in a conserved form, in which the phenomenological flux laws can be constructed in a number of alternative ways. We pay particular attention to flux laws that can account for non-locality through space fractional derivative operators. The available results on the well posedness of the governing equations using such flux laws are discussed. A discrete control volume numerical solution of the general conserved governing equation is developed and a general discrete treatment of boundary conditions, independent of the particular choice of flux law, is presented. The numerical properties of the scheme resulting from the flux laws are analyzed. We use numerical solutions of various test problems to compare the operation and predictive ability of two discrete fractional diffusion flux laws based on the Caputo (C) and Riemann–Liouville (RL) derivatives respectively. When compared with the C flux-law we note that the RL flux law includes an additional term, that, in a phenomenological sense, acts as an apparent advection transport. Through our test solutions we show that, when compared to the performance of the C flux-law, this extra term can lead to RL-flux law predictions that may be physically and mathematically unsound. We conclude, by proposing a parsimonious definition for a fractional derivative based flux law that removes the ambiguities associated with the selection between non-local flux laws based on the RL and C fractional derivatives.

我们从以守恒形式编写的扩散传输的一般控制方程开始，在该方程中，现象通量定律可以通过多种替代方式来构建。我们特别注意通过空间分数导数算子可以解释非局部性的通量定律。讨论了使用这种通量定律的控制方程的适定性的可用结果。提出了一般守恒控制方程的离散控制量数值解，给出了边界条件的一般离散处理，而与通量定律的具体选择无关。分析了由通量定律得出的方案的数值特性。我们使用各种测试问题的数值解来比较分别基于Caputo（C）和Riemann-Liouville（RL）导数的两个离散分数扩散通量定律的运算和预测能力。当与C通量定律进行比较时，我们注意到RL通量定律包括一个附加术语，从现象学的意义上讲，它充当明显的对流输运。通过我们的测试解决方案，我们发现，与C通量定律的性能相比，这个额外的项可能导致RL通量定律的预测在物理和数学上都不可靠。通过提出基于分数导数的通量定律的简约定义，可以得出结论，该定义消除了与基于RL和C分数导数的非局部通量定律之间的选择相关的歧义。