Windowed Fourier Methods for Overlapping Domain Approximations

用于重叠域近似的加窗傅里叶方法

• 批准号: 1522639
• 负责人: Rodrigo Platte
• 金额: $ 15万
• 依托单位: Arizona State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522639&HistoricalAwards=false
• 关键词: Windowed; Fourier; Methods; Overlapping; Domain

Partial differential equations on surfaces arise in many applications, including atmospheric research, evolution of coat patterns in animals, finger print formation, and deformation of elastic solids in fluids, to mention just a few. This research project aims to develop computationally efficient and accurate methods for numerical solution of such equations. The project is driven in particular by problems stemming from atmospheric models and will advance research on novel numerical schemes that are based on windowed Fourier approximations. Objectives include the design of algorithms for solving time-dependent differential equations on spherical geometries and other surfaces, the development of a rigorous mathematical analysis of convergence and stability for such algorithms, the implementation of high performance and scalable solvers, and to make efficient, accurate software available for use by the scientific community.Windowed Fourier methods are suitable for adaptive and parallel implementations and large-scale computations. The proposed schemes rely on domain decomposition, such as in the cubed sphere, with computations being carried out using fast Fourier transforms. Approximations are obtained on overlapping domains and a global solution is obtained by weighted average. Current high-order and spectral methods used in atmospheric research, such as spectral elements, are constrained by the well-known CFL condition; that is, node clustering in the space domain restricts the discretization in the time domain. Windowed Fourier methods use nearly uniform nodes on each subdomain, allowing for larger time steps when explicit schemes are used for time integration. For large-scale simulations, such as those required in climate prediction for example, each time step incurs significant communication cost at scale. By leveraging better node distributions, larger time steps and spectral accuracy, windowed Fourier schemes are expected to be more efficient than methods currently used in critical applications such as climate and geodynamics.

表面上的偏微分方程出现在许多应用中，包括大气研究，动物的皮毛图案的演变，指纹的形成，以及流体中弹性固体的变形，仅举几例。本研究项目的目的是开发计算效率和精确的方法，数值求解这些方程。该项目特别是由大气模型产生的问题驱动的，并将推进对基于窗口傅立叶近似的新数值方案的研究。目标包括设计用于求解球形几何形状和其他表面上的时间相关微分方程的算法，对这些算法的收敛性和稳定性进行严格的数学分析，实现高性能和可扩展的求解器，并使高效，精确的软件可供科学界使用。窗口傅立叶方法适用于自适应和并行实现，规模计算。所提出的方案依赖于区域分解，例如在立方球体中，使用快速傅里叶变换进行计算。在重叠区域上得到近似解，并通过加权平均得到全局解。目前在大气研究中使用的高阶和谱方法，如谱元素，受到众所周知的CFL条件的约束;即，空间域中的节点聚类限制了时域中的离散化。窗口傅立叶方法在每个子域上使用几乎均匀的节点，当显式方案用于时间积分时，允许更大的时间步长。对于大规模模拟，例如气候预测所需的模拟，每个时间步长都会产生大量的通信成本。通过利用更好的节点分布，更大的时间步长和光谱精度，窗口傅立叶方案预计将比目前在气候和地球动力学等关键应用中使用的方法更有效。