Adaptive solution of differential equations

微分方程的自适应求解

• 批准号: 8781-2007
• 负责人: Russell, Robert
• 金额: $ 5.68万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=389832
• 关键词: Adaptive; solution; differential; equations

Development of increasingly powerful supercomputers is primarily driven by the need to numerically solve nonlinear partial differential equations (PDEs), the essence of most complex mathematical models in science and engineering. A wide array of sophisticated software tools are required, perhaps the most challenging, both from a theoretical and practical point of view, being robust adaptive algorithms. The role of such an algorithm is deceptively simple to state: recognize the regions where a physical solution is difficult to approximate, provide some measure of the extent of that difficulty, and adapt the numerical solution process accordingly. While scientists and engineers doing large scale computation have out of necessity long used adaptive algorithms, scientific computing as a discipline was initially slow to embrace its fundamental theoretical importance. Nevertheless, today journals are replete with papers analyzing adaptivity. Over the last fifteen years we have developed a class of adaptive techniques called moving mesh methods, which despite being slow to gain acceptance, are now viewed by increasing numbers of experts as an indispensible component of general numerical PDE algorithms for solving complex problems in areas as diverse as climate modelling and engineering design. Our research, focussed now mainly on higher dimensional problems, is proceeding apace on several fronts: (1) Most of the classical adaptive algorithms have traditionally been based upon variational techniques. Inspired by early work of engineers, we are investigating a fascinating new class of algorithms motivated more by geometric principles. (2) The acid test of any method being its ability to solve real world problems, we are refining a variety of techniques and developing mathematical software which incorporates the different moving mesh approaches. (3) We are applying these adaptive algorithms in new ways such as for moving front problems, to other classes of PDEs such as 4th order problems capable of modelling rich solution behaviour, and to new areas of application such as image registration. (4) Lastly, our ultimate goal is to provide a unifying framework and theory for mesh adaptivity.

越来越强大的超级计算机的发展主要是由数值求解非线性偏微分方程（PDE）的需求驱动的，这是科学和工程中最复杂的数学模型的本质。一系列复杂的软件工具是必要的，也许是最具挑战性的，无论是从理论和实践的角度来看，是强大的自适应算法。这种算法的作用看似简单：识别物理解难以近似的区域，提供困难程度的某种度量，并相应地调整数值求解过程。虽然进行大规模计算的科学家和工程师长期以来一直不必要地使用自适应算法，但科学计算作为一门学科最初在接受其基本理论重要性方面进展缓慢。然而，今天的期刊上充斥着分析适应性的论文。在过去的15年中，我们已经开发了一类自适应技术称为移动网格方法，尽管获得认可的速度很慢，但现在越来越多的专家将其视为解决气候建模和工程设计等领域复杂问题的通用数值PDE算法的不可或缺的组成部分。我们的研究，现在主要集中在高维问题上，正在进行快速在几个方面：（1）大多数经典的自适应算法传统上是基于变分技术。受工程师早期工作的启发，我们正在研究一类更受几何原理启发的迷人的新算法。(2)任何方法的严峻考验都是其解决真实的世界问题的能力，我们正在改进各种技术并开发包含不同移动网格方法的数学软件。(3)我们正在应用这些自适应算法的新方法，如移动前的问题，其他类的偏微分方程，如4阶问题，能够模拟丰富的解决方案的行为，以及新的应用领域，如图像配准。(4)最后，我们的最终目标是为网格自适应提供一个统一的框架和理论。