Windowed Fourier Methods for Overlapping Domain Approximations

表面上的偏微分方程在许多应用中出现，包括大气研究，动物皮毛图案的演变，指纹的形成，以及流体中弹性固体的变形，仅举几例。本研究项目旨在为此类方程的数值解提供高效且精确的计算方法。该项目特别受到大气模型问题的推动，并将推进基于窗口傅立叶近似的新型数值方案的研究。目标包括设计用于在球面几何和其他表面上求解随时间变化的微分方程的算法，对此类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，并为科学界提供高效，准确的软件。窗口傅里叶方法适用于自适应并行实现和大规模计算。所提出的方案依赖于区域分解，例如在立方球中，使用快速傅里叶变换进行计算。在重叠域上得到近似，并通过加权平均得到全局解。目前用于大气研究的高阶和光谱方法，如光谱元素，受到众所周知的CFL条件的限制；即空间域的节点聚类限制了时间域的离散化。窗口傅里叶方法在每个子域上使用几乎一致的节点，当使用显式方案进行时间积分时，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规模的通信成本。通过利用更好的节点分布、更大的时间步长和光谱精度，窗口傅里叶方案有望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。

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.