A Finite Difference Approach to Pseudospectral Methods

伪谱（PS）方法是有限差分（FD）和有限元（FEM）方法的高精度替代方法。它们对于长时间求解对流占优方程和相对简单的几何形状特别有效。无论是算法本身和他们的分析传统上都被封闭地绑在不同类的正交functions.The最近的书“一个实用指南伪谱方法”（由thempresent调查员）指出，大量的推广，增强和见解可以通过查看PS方法，而不是FD方法的特殊情况下获得。在DMS-9706916赠款下探索了几个这样的机会。在这些经验的基础上，我们现在将引入径向基函数（RBFs）作为与PS方案集成的附加组件。虽然计算成本很高，但径向基函数的光谱精度，加上它们极端的几何灵活性，似乎使它们非常适合在不规则边界附近补充PS方法。为了实现这一点，需要大量的研究，包括RBF近似的基本特征，以及与PS和FD方法接口的相关问题的研究。伪谱（PS）方法最早是在20世纪70年代初提出的（与气象学和湍流模拟有关）。他们已经被证明是非常有效的高精度计算在许多其他领域，如计算电磁学和非线性波。具有战略意义的重要应用领域包括模拟飞机的雷达散射、计算机芯片组件之间的电干扰以及光纤中的信号传输。PS方法的主要缺点是难以应用于复杂的几何形状。其中一些问题在最近几年已经被克服了（通常是通过将复杂的域分解为较简单的部分，然后应用一些合适的计算方法来耦合子域）。一种完全不同的、非常有前途的方法将在该资助下被介绍和探索。这涉及到结合“经典”PS方法与几何上非常灵活的方法称为径向基函数（RBFs）。这从来没有人尝试过，研究福尔斯属于“高风险，高收益”的范畴。成功是不确定的，但如果发生，它可能会产生重大影响，特别是在计算电磁学的应用，如模拟雷达散射/隐身性能的对象。

Pseudospectral (PS) methods are high-accuracy alternatives to finitedifference (FD) and finite element methods (FEM) for the numerical solutionof PDEs. They are particularly effective for solving convection-dominatedequations over long times and in relatively simple geometries. Both thealgorithms themselves and the analysis of them have traditionally beenclosely tied to expansions in different classes of orthogonal functions.The recent book "A Practical Guide to Pseudospectral Methods" (by thepresent investigator) notes that a large body of generalizations,enhancements and insights can be gained by viewing PS methods instead asspecial cases of FD methods. Several such opportunities were explored underthe grant DMS-9706916. Building on these experiences, we will now introduceradial basis functions (RBFs) as an additional component to be integratedwith PS schemes. Although computationally quite costly, the spectralaccuracy of RBFs, combined with their extreme geometric flexibility, wouldseem to make them ideally suited to complement PS methods in the vicinityof irregular boundaries. To realize this requires much research regardingthe basic features of RBF approximations, as well as research on issuesrelated to interfacing them with PS and FD methods. Pseudospectral (PS) methods were first proposed in the early 1970's (inconnection with meteorology and turbulence modeling). They have since beenshown to be extraordinary effective for high-accuracy calculations innumerous other fields, such as computational electromagnetics and nonlinearwaves. Strategically important application areas include simulating theradar scattering from airplanes, the electrical interference betweencomponents on computer chips, and the transmissions of signals in opticalfibres. The main weakness with PS methods has been the difficulty to applythem in complex geometries. Some of this has been overcome in recent years(usually by splitting a complicated domain in simpler parts, and thenapplying some suitable computational means for coupling of the subdomains).A quite different and highly promising approach will be introduced and thenexplored under this grant. This involves combining 'classical' PS methodswith the geometrically extremely flexible approach known as radial basisfunctions (RBFs). This has never been attempted, and the research falls inthe 'high risk, high gain' category. Success is by no means certain, but ifit happens, it could have a major impact, particularly on computationalelectromagnetics in applications such as simulation of radar scattering /stealth properties of objects.