Analysis of advanced discretizations of partial differential equations

偏微分方程的高级离散化分析

• 批准号: RGPIN-2015-05733
• 负责人: Tsogtgerel, Gantumur
• 金额: $ 1.82万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613717
• 关键词: Analysis; advanced; discretizations; partial; differential; Analysis; advanced; discretizations; partial; differential

Modelling and simulation using partial differential equations are ubiquitous in science and engineering, including astronomy, meteorology, oceanography, seismology, geophysics, geology, economics, fluid mechanics, solid state physics, and quantum mechanics. In the last few decades we have seen an enormous increase in the complexity of a typical simulation that can be run on a computer. One reason is of course the amazing technological leaps in computer hardware we have been witnessing. On the other hand, perhaps surprisingly, it is estimated that the development of fast algorithms has had roughly the same impact as that of the hardware improvements. In fact, we expect much more: The existing fast algorithms and their theory can be compared to the tip of an iceberg, with most of the treasures yet to be discovered. The theme of the proposed project is the theoretical understanding of two important classes of numerical algorithms for solving partial differential equations: Adaptive methods and geometric discretization techniques. While adaptive methods can be described as algorithms that distribute computing resources in a "smartest" way so as to minimize waste of effort, the goal of geometric discretization techniques is to preserve fundamental geometric properties of the original differential equations. The latter techniques are known to be generally preferable to the more conventional methods, and in many cases, such as simulations of bio-chemical molecules, they are the only reasonable choice. The current project aims to further mathematical understanding of certain numerical algorithms belonging to either or both of the aforementioned classes. First, we plan to design new fast algorithms for solving various equations that are used to model bio-membranes, large scale ocean flows, and electromagnetic phenomena. Second, we will perform rigorous mathematical analysis of the designed algorithms, building a comprehensive theory that explains how the algorithms behave in different situations. This is important since no algorithm is completely fool proof, and depending on the particular problem at hand, one might want to choose different algorithms. The third aim of the project is mathematical analysis of some existing geometric discretization methods that are used in simulations of sub-nucleonic matter and of violent astronomical events such as black hole collisions. These methods work reasonably well in practice, but not without difficult unresolved problems, and currently we have a very little understanding of them from mathematical standpoint. We expect that rigorous mathematical treatment will not only give more confidence to the practitioners, but also open up possibilities to resolve the issues and improve upon the existing algorithms.

使用部分微分方程的建模和模拟在科学和工程中无处不在，包括天文学，气象，海洋学，地震学，地球物理学，地质学，经济学，经济学，流体力学，固态物理学和量子力学。在过去的几十年中，我们发现可以在计算机上运行的典型模拟的复杂性有所提高。当然，一个原因是我们一直在目睹的计算机硬件中惊人的技术飞跃。另一方面，也许令人惊讶的是，据估计，快速算法的开发与硬件改进的影响大致相同。实际上，我们期望更多：现有的快速算法及其理论可以与冰山一角进行比较，其中大多数宝藏尚待发现。拟议项目的主题是对解决部分微分方程的两种重要类数算法的理论理解：自适应方法和几何离散技术。虽然自适应方法可以描述为以“最聪明”的方式分发计算资源以最大程度地浪费努力的算法，但几何离散技术的目的是保留原始微分方程的基本几何特性。已知后来的技术通常比更传统的方法更喜欢，在许多情况下，例如生物化学分子的模拟，它们是唯一合理的选择。当前的项目旨在进一步对属于或两个相关类别的某些数值算法的数学理解。首先，我们计划设计新的快速算法，以求解用于建模生物膜，大规模洋流和电磁现象的各种方程。其次，我们将对设计算法进行严格的数学分析，并建立一个综合理论，该理论解释了算法在不同情况下的行为方式。这很重要，因为没有算法完全万无一失，并且根据手头的特定问题，人们可能想选择不同的算法。该项目的第三个目的是对一些现有的几何离散方法的数学分析，这些方法用于模拟亚核物质和暴力天文事件（例如黑洞碰撞）。这些方法在实践中运作良好，但并非没有困难的未解决问题，目前，从数学角度来看，我们对它们的理解很少。我们预计，严格的数学处理不仅会给实践者提供更多的信心，而且还为解决问题并改善现有算法的可能性开辟了可能性。