Advancements in the Ultraspherical Spectral Method

超球面光谱方法的进展

• 批准号: 1522577
• 负责人: Alex Townsend
• 金额: $ 14.48万
• 依托单位: Massachusetts Institute of Technology
• 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: 2015-08-01 至 2016-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522577&HistoricalAwards=false
• 关键词: Advancements; Ultraspherical; Spectral; Method

The numerical solution of real-world fluid flow and airfoil problems needs an accurate, flexible, and fully-adaptive spectral element method. The so-called ultraspherical spectral method, with its sparsity and regularity preserving discretizations, is promising to overcome many of the traditional computational barriers. This research project will exploit and investigate the remarkable properties of the ultraspherical spectral method with the aim of producing a high quality and industrial-strength spectral element solver for partial differential equations. One key feature will be its robustness to pinching boundary features, typical with airfoils, that will alleviate the current tremendous burden on mesh generation algorithms. The project will radically alter the perception of spectral methods in the computational mathematics and engineering communities by extensively demonstrating that, when done carefully, they can be a flexible, general, and powerful numerical tool.Today's pseudospectral methods deliver both convenience and spectrally accurate discretizations for the solution of differential equations. However, they lead to dense discretizations, numerical instability, and a severe limitation to simple geometries. The novel ultraspherical spectral method is an alternative that retains the same accuracy and convenience, but leads to almost banded well-conditioned discretizations that faithfully preserves the regularity of the underlying differential operator while also being amenable to specialized fast linear algebra routines. Based on this new spectral method, the PI will derive a new mathematically-grounded fully-adaptive spectral element method for meshed geometries. Key novel computational features will include: (1) A high accuracy on mesh elements that is independent of the aspect ratio; (2) True hp-adaptivity that allows for essentially arbitrarily large element degree p and small average mesh element size h (without concern of ill-conditioning); and (3) The flexibility to solve a wide range of differential equations with general boundary constraints; and (4) Local refinement and mesh coarsening for the resolution of corner singularities. This new spectral element method will be applied to challenging partial differential equations for the state-of-the-art numerical simulation of advection-dominated fluid flow problems.

现实世界流体流和机翼问题的数值解决方案需要准确，灵活和完全自适应的光谱元素方法。所谓的超球光谱法具有稀疏性和规律性保留离散性，有望克服许多传统的计算障碍。该研究项目将利用和研究超球光谱方法的显着特性，目的是为部分微分方程生产高质量和工业强度的光谱元素求解器。一个关键特征将是它对捏合边界特征的稳健性（典型的带有机翼），这将减轻当前对网格生成算法的巨大负担。该项目将从根本上改变计算数学和工程社区中光谱方法的感知，从而广泛地证明，当仔细完成时，它们可以是一种灵活，一般且功能强大的数值工具。Today的伪谱方法提供了便利性和光谱准确的差异方程解决方案的确定性。但是，它们导致了密集的离散性，数值不稳定以及对简单几何形状的严重限制。新型的超球光谱方法是一种替代方法，可保留相同的准确性和便利性，但导致几乎带有良好条件的离散化，忠实地保留了基础差分运算符的规律性，同时也可以与专用的快速线性代数常规固定。基于这种新的光谱方法，PI将用于网状几何形状的新的数学基础完全自适应光谱元素方法。关键的新型计算特征将包括：（1）与纵横比无关的网格元素的高精度； （2）真正的HP适应性允许基本上任意较大的元素程度P和较小的平均网格元素尺寸H（不关心不良条件）； （3）求解具有一般边界约束的各种微分方程的灵活性； （4）局部改进和网状，以解决角落奇点的分辨率。这种新的频谱元素方法将应用于挑战的部分微分方程，以对以对流为主的流体流量问题进行最新的数值模拟。