A Finite Difference Approach to Pseudospectral Methods

伪谱方法的有限差分法

• 批准号: 9706916
• 负责人: Bengt Fornberg
• 金额: $ 8.1万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706916&HistoricalAwards=false
• 关键词: Finite; Difference; Approach; Pseudospectral; Methods

9706916 B. Fornberg Pseudospectral (PS) methods - a high-accuracy alternative to finite difference (FD) and finite elemnt methods - are particularly effective for solving convection-dominated PDEs over long times and in relatively simple geometries. Both the algorithms themselves and their analysis have usually been closely tied to expansions in different classes of orthogonal functions. The recent book "A Practical Guide to Pseudospectral Methods" (by the present investigator) notes that a large body of generalizations, enhancements and insights can be gained by viewing PS methods instead as special cases of FD methods. These opportunities will now be explored further, in particular with the aim of combining the PS approach with domain decomposition for time-dependent computational electromagnetics. Pseudospectral (PS) methods were first proposed in the early 1970's (in connection with meteorology and turbulence modeling). They are now routinely used in many fields such as nonlinear wave motions, weather forecasting, fluid mechanics, computational chemistry, and with time-domain computational electromagnetics (TD CEM) emerging as a new and potentially major application area. This present investigation will explore how a series of new leads for PS methods can be developed further and then brought to bear on applications, in particular in the field of TD CEM. Such applications include simulations of the radar visibility of objects and of how electromagnetic fields are generated by/ interact with components on integrated circuits.

9706916 B. Fornberg 伪谱 (PS) 方法是有限差分 (FD) 和有限元方法的高精度替代方法，对于长时间求解相对简单的几何结构中的对流主导的偏微分方程特别有效。算法本身及其分析通常与不同类别的正交函数的展开密切相关。最近的书《伪谱方法实用指南》（由本研究者撰写）指出，通过将 PS 方法视为 FD 方法的特例，可以获得大量的概括、增强和见解。 现在将进一步探索这些机会，特别是为了将 PS 方法与瞬态计算电磁学的域分解相结合。 伪光谱 (PS) 方法于 20 世纪 70 年代初首次提出（与气象学和湍流建模相关）。它们现在通常用于许多领域，例如非线性波动、天气预报、流体力学、计算化学，并且时域计算电磁学 (TD CEM) 正在成为一个新的潜在主要应用领域。本研究将探索如何进一步开发 PS 方法的一系列新线索，然后应用于应用，特别是在 TD CEM 领域。此类应用包括模拟物体的雷达可见度以及电磁场如何由集成电路上的组件生成/与集成电路上的组件相互作用。