Development and analysis of fast numerical methods for fractional diffusion and advection-diffusion equations

分数扩散方程提供了表现出异常扩散的输运过程的充分描述，这是经典二阶扩散方程不能正确建模的。然而，分数扩散方程引入了严重的计算、数值和数学上的困难，这在二阶方程的背景下是没有遇到的：(i)分数扩散方程导致具有密集或全系数矩阵的数值方法，这使得现实的三维模拟在计算上难以处理！（ii）分数阶扩散算子是非局部的，分数阶微分算子的伴随算子本身不是负数，这使得数学分析变得非常复杂。本建议的目标如下：(i)发展分数扩散方程的快速数值方法，在保持标准方法的稳定性和准确性的同时，大大提高了计算效率和内存要求。（ii）为快速数值方法开发有效的预条件，使预条件线性系统的收敛与网格大小无关。（iii）对提出的快速方法进行相应的数学和数值分析。扩散过程无处不在，发生在自然、科学、社会科学和工程中。样本应用包括水和营养物质如何在生物体中穿过膜，蚊子如何传播疟疾，复印机和激光打印机如何工作，地下水中的污染物如何运输，以及生物细胞的信号传导，动物的觅食行为和金融。菲克在1855年首次建立了扩散方程。但正是爱因斯坦从第一原理推导出了扩散方程，作为他研究布朗运动的一部分。近几十年来，人们发现越来越多的扩散过程不能用经典扩散方程来恰当地模拟。这些发现具有深远的影响。例如，最近由分数平流扩散方程建立的模型表明，污染含水层的修复可能比以前由经典平流扩散方程预测的时间长几十年或几个世纪。因此，进一步的调查至关重要。这项工作的结果将适用于广泛的应用。拟议的研究活动还将为研究生和本科生提供先进的跨学科培训。所有这些活动都将产生广泛而持久的影响，并直接促进国家的智力基础设施。

Fractional diffusion equations provide an adequate description of transport processes that exhibit anomalous diffusion, which cannot be modeled properly by classical second-order diffusion equations. However, fractional diffusion equations introduce severe computational, numerical, and mathematical difficulties which have not been encountered in the context of second-order equations: (i) Fractional diffusion equations lead to numerical methods with dense or full coefficient matrices, which makes realistic three-dimensional simulations computationally intractable! (ii) Fractional diffusion operators are non-local and the adjoint of a fractional differential operator is not the negative of itself, which significantly complicates the mathematical analysis. The objectives of this proposal are as follows: (i) Develop fast numerical methods for fractional diffusion equations with significantly improved computational efficiency and memory requirement while retaining the stability and accuracy of standard methods. (ii) Develop efficient preconditioners for the fast numerical methods, so that the convergence of the preconditioned linear system is independent of the mesh size. (iii) Conduct corresponding mathematical and numerical analysis for the proposed fast methods.Diffusion processes are ubiquitous and occur in nature, sciences, social sciences, and engineering. Sample applications include how water and nutrients travel through membranes in living organisms, how mosquitoes spread malaria, how copiers and laser printers work, and how contaminants in groundwater are transported, as well as the signaling of biological cells, foraging behavior of animals, and finance. Fick first sat up the diffusion equation in 1855. But it was Einstein who derived the diffusion equation from first principle as part of his work on Brownian motion. In last few decades it was found that increasingly more diffusion processes cannot be properly modeled by classical diffusion equations. These discoveries have profound consequences. For example, recent modeling by fractional advection-diffusion equations indicate that remediation of contaminated aquifers may take decades or centuries longer than previously predicted by the classical advection-diffusion equations. Hence, further investigations are crucial. The results of this work will be applicable to a wide range of applications. The proposed research activities will also provide advanced interdisciplinary training to graduate and undergraduate students. All of these activities will have broad and long-lasting impacts and contribute directly to the intellectual infrastructure of the nation.