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

分数扩散和平流扩散方程快速数值方法的开发和分析

• 批准号: 1216923
• 负责人: Hong Wang
• 金额: $ 24万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1216923&HistoricalAwards=false
• 关键词: Development; analysis; fast; numerical; methods

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.

分数扩散方程充分描述了表现出反常扩散的输运过程，而反常扩散无法通过经典的二阶扩散方程进行正确建模。然而，分数扩散方程引入了二阶方程中未遇到的严重的计算、数值和数学困难：（i）分数扩散方程导致具有密集或完整系数矩阵的数值方法，这使得现实的三维模拟在计算上变得困难！ (ii) 分数扩散算子是非局部的，并且分数阶微分算子的伴随物不是其自身的负数，这使得数学分析变得非常复杂。该提案的目标如下：（i）开发分数扩散方程的快速数值方法，显着提高计算效率和内存需求，同时保留标准方法的稳定性和准确性。 (ii) 为快速数值方法开发有效的预处理器，以便预处理线性系统的收敛与网格尺寸无关。 (iii) 对所提出的快速方法进行相应的数学和数值分析。扩散过程普遍存在于自然、科学、社会科学和工程领域。示例应用包括水和营养物质如何穿过生物体的膜、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传输，以及生物细胞的信号传导、动物的觅食行为和金融。菲克于 1855 年首次提出扩散方程。但爱因斯坦根据第一原理推导了扩散方程，作为其布朗运动研究的一部分。在过去的几十年中，人们发现越来越多的扩散过程无法通过经典扩散方程正确建模。这些发现具有深远的影响。例如，最近通过分数平流扩散方程进行的建模表明，污染含水层的修复可能需要比经典平流扩散方程先前预测的时间长数十年或数百年的时间。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，并直接为国家的智力基础设施做出贡献。