Advancements in the Ultraspherical Spectral Method

真实世界流体流动和翼型问题的数值求解需要一种精确、灵活和完全自适应的谱单元方法。所谓超球谱方法，由于其稀疏性和正则性保持离散化，有望克服许多传统的计算障碍。这项研究项目将开发和研究超球谱方法的显著特性，目的是产生一个高质量和工业强度的偏微分方程组的谱元素求解器。其中一个关键特征是它对挤压边界特征的稳健性，这是翼型的典型特征，这将减轻目前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界对谱方法的看法，因为它广泛地展示了，如果仔细操作，它们可以成为一种灵活、通用和强大的数值工具。今天的伪谱方法为微分方程组的求解提供了方便和频谱准确的离散化。然而，它们会导致密集的离散化、数值不稳定以及对简单几何图形的严重限制。新的超球谱方法是一种替代方法，它保持了相同的精度和便利性，但导致了几乎带状的良好条件离散化，忠实地保持了基本微分算子的正则性，同时也服从专门的快速线性代数例程。基于这种新的谱方法，PI将推导出一种新的网格几何的数学基础的全自适应谱单元方法。主要的新颖计算功能将包括：(1)与长宽比无关的网格单元的高精度；(2)真正的hp-自适应性，允许基本上任意大的单元阶数p和小的平均网格单元尺寸h(无需考虑病态)；(3)灵活地求解具有一般边界约束的各种微分方程组；以及(4)用于解决角点奇异性的局部细化和网格粗化。这种新的谱元素方法将被用于挑战偏微分方程组，用于最新的对流主导流体流动问题的数值模拟。

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.