The fractional dispersion advection equations (FDAEs) have recently attracted considerable attention due to their extensive application in the fields of science and engineering. For example, it has been shown that the anomalous solute transport behaviour that exists in hydrology can be well explained by introducing FDAEs. Therefore, the study of FDAEs has profound significance for understanding real transport phenomena in nature. Nevertheless, the existing algorithms for the FDAEs are generally intricate and costly. Therefore, exploiting an efficient solution technique has been a concern for scientists. In an effort to overcome this challenge, a promising lattice Boltzmann (LB) model for the FDAEs is presented in this paper. The Riemann–Liouville definition and the Grünwald–Letnikov definition are introduced for the time derivatives. In addition, Chapman–Enskog analysis is applied to recover the FDAEs. To test the validity of the model, three numerical examples are carried out. In addition, a comparative study of the proposed model and the classical implicit finite difference scheme is also conducted. The numerical results show that the model is suitable for simulating FDAEs.

中文翻译：

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基于Lattice Boltzmann模型的分数色散对流方程的数值模拟

由于分数弥散对流方程（FDAE）在科学和工程领域的广泛应用，最近引起了相当大的关注。例如，已经证明，引入FDAE可以很好地解释水文学中存在的异常溶质运移行为。因此，对FDAEs的研究对于理解自然界中的真实运输现象具有深远的意义。然而，用于FDAE的现有算法通常是复杂且昂贵的。因此，开发有效的解决方案技术一直是科学家关注的问题。为了克服这一挑战，本文提出了一种有前途的FDAE格子Boltzmann（LB）模型。为时间导数引入了黎曼-利维尔定义和格伦瓦尔德-列特尼科夫定义。此外，Chapman–Enskog分析用于回收FDAE。为了测试模型的有效性，进行了三个数值示例。此外，还对提出的模型与经典隐式有限差分方案进行了比较研究。数值结果表明该模型适合于模拟FDAE。