分数阶对流扩散方程的格子Boltzmann方法研究

Anomalous diffusion phenomenon has been widely used in groundwater contaminant transport studying. Fractional order advection-diffusion equation is a powerful tool for solving the problem. How to design a numerical algorithm with high efficiency and accuracy, is one of the hot topics that are highly concerned in the areas of science and engineering. The project attempts to study the numerical schemes for time-fractional advection-diffusion in Caputo sense by the lattice Boltzmann method which is mesoscopic kinetic method, including: studying on how to approximate the fractional order derivative; for the fractional advection sub-diffusion problem, deriving the single-relaxation-time and multi-relaxation-time LB models by multi-scale expansion method; for fractional advection super-diffusion, deriving the LB model for hyperbolic equation which is mainly concerned to be solved; for the anomalous diffusion problems coupled with the flow field and solute field, proposing double distribution single-relaxation-time and multi-relaxation-time LB models and carrying on theoretical analysis and numerical simulation; in view of the locality of LB method, exploring high efficient algorithm to build GPU computing platform based on CUDA to solve large amount of storage and computation, and extending these models to three dimensional problem with optimization algorithm. The research of the project will promote the modeling and application of LB method on fractional advection-diffusion equations, and promote the development of environmental mechanics, computational fluid mechanics and other related research fields.

反常扩散现象广泛存在于污染物在地下水中的迁移过程，分数阶对流扩散方程是求解该问题的有力建模工具，如何设计高效快速的数值算法是科学与工程界高度关注的热点问题之一。本项目基于介观动力学的格子Boltzmann方法对Caputo型时间分数阶对流扩散方程开展数值模拟研究工作。研究内容包括：构建分数阶导数的近似公式；对于分数阶对流慢扩散问题，通过多尺度展开推导相应的单松弛和多松弛LB模型；对于分数阶对流超扩散问题，重点需要解决的是如何构建求解双曲型方程的LB模型；对于流场-溶质场耦合的反常扩散问题，建立双分布的单松弛和多松弛LB模型，并作理论分析和数值模拟计算；鉴于LB方法的局部性，探讨基于CUDA的GPU计算的高效算法，以解决计算量和存储量大的问题，并将该模型推广到三维，优化算法。本项目的完成将推动LB方法在分数阶对流扩散方程中的建模与应用，促进环境力学、计算流体力学等相关学科的发展。

反常扩散在研究污染物地下水中的迁移规律有着重要的应用，分数阶对流扩散方程是求解该问题的有力建模工具。本项目基于格子Boltzmann方法和有限差分方法围绕着反常扩散问题提出高效的数值计算格式。（1）对时间分数阶低扩散模型利用复合求积公式和L1插值，提出了求解该问题的格子Boltzmann模型，通过多尺度展开恢复宏观方程，数值实验验证理论分析结果；对流场-溶质场耦合的反常扩散问题，建立双分布LB模型，通过Chapman-Enskog分析恢复到宏观方程，数值实验说明了格式的有效性；（2）在时间分数阶低扩散的LB模型基础上，提出了求解分数阶波方程的格子Boltzmann模型，包括一维、二维和三维模型，并将该算法推广到分数阶Klein-Gordon equation方程和分数阶sine-Gordon equation方程；（3）对于上述模型推广到高维问题，基于指数核逼近和GPU并行计算解决了分数阶导数计算量、存储量大的问题，大大提高模型的计算效率，加速比最高达110，为后续实际应用奠定计算基础；（4）对多孔介质中两相流的LBGK建模，得到二维、三维LB模型，对于三维计算量大问题，开发了GPU并行程序；（5）提出了非线性分数阶高效有限差分方法、对初值具有弱正则性的反常扩散问题提出了分级网格的快速差分格式。.通过项目的实施，推动LB方法在分数阶微分方程中的建模与应用，促进环境力学、计算流体力学等相关学科的发展，为研究多尺度反常扩散问题提供良好的研究基础。