非均匀介质中tempered分数阶反常扩散模型的高效数值算法研究

Tempered anomalous diffusion describes the very slow transition from anomalous to normal diffusion and it has many applications in geophysics, finance, hydrology, and other fields. The project will develop corresponding fast numerical algorithms for several tempered fractional partial differential equations (TFPDEs) which simulate the solute transport model. Firstly, the high order numerical approximations are constructed for Caputo type and Riemann-Liouville type time-tempered fractional operators, and fast numerical algorithms of time-TFPDEs with non-smooth solutions are derived; Secondly, the high order numerical approximations are proposed for space-tempered fractional operators, and we design some fast numerical algorithms for space-time TFPDEs; Thirdly, by some new construction techniques and preconditioned strategy, we design some novel numerical algorithms including fast compact alternating direction implicit schemes and locally one-dimensional methods for multi-dimensional space-time-TFPDEs in the preconditioned method, and these resulting schemes can keep the accuracy of the order 2 in time. Last but not least, numerical analysis is established by a new technique of discrete energy analysis, and all numerical algorithms will be applied to practical problems. These results of the project will provide theory and algorithms for the solute transport model and related fields of engineering.

Tempered反常扩散用于描述从反常扩散到正常扩散的缓慢收敛过程，广泛应用于地球物理学、金融学和水文学等领域。本项目拟对模拟溶质迁移模型的几类tempered分数阶偏微分方程建立相应的高效数值算法。首先对Caputo型和Riemann-Liouville型时间tempered分数阶算子提出新型高阶数值逼近公式，进而对含非光滑解的时间tempered分数阶偏微分方程建立快速数值算法；其次，构造空间tempered分数阶算子的高阶逼近公式，对时空tempered分数阶偏微分方程建立快速数值算法，然后采用新的构造方法和预处理策略，对多维时空tempered分数阶偏微分方程建立2阶的新型预处理交替方向隐格式和局部一维格式等快速数值算法；最后采用新型能量分析法，系统地给出数值分析理论，并将相应的算法应用于实际问题。通过本项目的研究可以为溶质迁移模型及相关工程领域提供算法和理论保障。

近几年来，Tempered分数阶偏微分方程越来越多的用于描述自然介质中溶质迁移的弥散现象。本项目对几类模拟溶质迁移模型的tempered分数阶偏微分方程建立了相应的高效数值算法。首先对一类时间tempered分数阶可动/不动区变系数偏微分方程建立了高阶数值格式；其次对非线性时间分数阶可动/不可动区偏微分方程建立了线性化高阶数值格式；然后基于一种新型线性化技巧，对非线性空间分数阶对流弥散方程建立了两层线性化数值格式。最后对二维时空分数阶可动/不可动区次扩散方程建立了有效的高阶交替方向隐格式。本项目的研究可以为溶质迁移模型及相关工程领域提供算法和理论保障。