Mathematical Sciences: Computational Analysis of Multiple Scales Problems in Wave Propagation

数学科学：波传播中多尺度问题的计算分析

• 批准号: 9600146
• 负责人: Thomas Hagstrom
• 金额: $ 6万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9600146&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Computational; Analysis; Multiple

9600146 Hagstrom This project's primary focus is on the development, analysis and implementation of new and more efficient numerical methods for simulating wave propagation in the presence of multiple temporal and spatial scales. This includes the construction of efficient discretization methods and accurate radiation conditions for solving linear hyperbolic systems, the development of adaptive spectral methods for problems requiring highly accurate solutions, and the analysis of splitting schemes for the efficient integration of multiple time scales problems for partial differential equations. Specific physical applications to be considered include acoustic and electromagnetic wave propagation and the challenging problem of simulating dynamic combustion phenomena making use of detailed models of the physics. Basic analyses of the governing systems of nonlinear partial differential equations will also be undertaken. Multiple scales problems are difficult because the uniform resolution of the smallest scales present would lead to a prohibitively large computational problem, even for the fastest machines likely to be available in the coming decades. They are also a common feature of scientifically and technologically important phenomena in a number of distinct physical settings. They arise sometimes from the need to solve problems on large spatial domains or over long time periods, and other times from the appearance of very localized solution features, such as sharp wavefronts. In the latter case they are generally associated with singular perturbations of the governing equations. Singular perturbations may be thought of as small changes in the equations which can lead to large qualitative and quantitative changes in the solutions. Such problems can also often be analyzed by non- numerical asymptotic techniques and in this work asymptotic analysis is used in a number of ways to guide and complement numerical computations. The translation of theoretical advances into useful tools, and the consideration of more complicated and comprehensive mathematical models is emphasized so as to maximize the impact of the research on other science and engineering disciplines.

9600146 Hagstrom 该项目的主要重点是开发、分析和实施新的、更有效的数值方法，用于模拟存在多个时间和空间尺度的波传播。 这包括构建高效的离散化方法和精确的辐射条件来求解线性双曲系统，开发自适应谱方法来解决需要高精度解决的问题，以及分析分裂方案以有效积分偏微分方程的多个时间尺度问题。要考虑的具体物理应用包括声波和电磁波传播以及利用详细的物理模型模拟动态燃烧现象的挑战性问题。还将对非线性偏微分方程的控制系统进行基本分析。 多尺度问题很困难，因为即使对于未来几十年可能出现的最快的机器，最小尺度的统一分辨率也会导致过于庞大的计算问题。它们也是许多不同物理环境中具有重要科学和技术意义的现象的共同特征。它们有时是由于需要解决大空间域或长时间段上的问题而产生的，有时是由于非常局部化的解决方案特征（例如尖锐的波前）的出现而产生的。在后一种情况下，它们通常与控制方程的奇异扰动相关。奇异扰动可以被​​认为是方程中的微小变化，这可能导致解的巨大的定性和定量变化。此类问题通常也可以通过非数值渐近技术进行分析，并且在这项工作中，渐近分析以多种方式用于指导和补充数值计算。强调将理论进步转化为有用的工具，并考虑更复杂和更全面的数学模型，以最大限度地发挥研究对其他科学和工程学科的影响。