A Finite Difference Approach to Pseudospectral Methods

伪谱法（PS）是替代有限差分法（FD）和有限元法（FEM）求解偏微分方程的高精度方法。它们对于求解长时间和相对简单的几何形状的对流主导方程特别有效。传统上，算法本身和它们的分析都与不同类别的正交函数的展开紧密联系在一起。最近的书“伪谱方法实用指南”（由现任研究者撰写）指出，通过将PS方法视为FD方法的特殊情况，可以获得大量的概括，增强和见解。在DMS-9706916的资助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（rbf）作为与PS方案集成的附加组件。虽然计算成本相当高，但rbf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，需要对RBF近似的基本特征进行大量研究，以及研究与PS和FD方法相结合的相关问题。伪谱（PS）方法是在20世纪70年代初首次提出的（与气象学和湍流模拟有关）。从那以后，它们在许多其他领域的高精度计算中被证明是非常有效的，比如计算电磁学和非线性波。具有重要战略意义的应用领域包括模拟飞机上的雷达散射，计算机芯片上组件之间的电子干扰，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。近年来已经克服了其中的一些问题（通常是将复杂的领域分成更简单的部分，然后应用一些合适的计算方法来耦合子领域）。在这项资助下，将引入并探索一种完全不同的、非常有前途的方法。这涉及到将“经典”PS方法与几何上极其灵活的称为径向基函数（rbf）的方法相结合。这从未被尝试过，而且这项研究属于“高风险、高收益”的范畴。成功绝不是确定的，但如果成功了，它可能会产生重大影响，特别是在计算电磁学应用中，如模拟雷达散射/物体隐身特性。

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.