A Finite Difference Approach to Pseudospectral Methods

伪谱方法是数值求解偏微分方程组的有限差分方法和有限元方法的高精度替代方法。它们对于在较长时间内和在相对简单的几何中求解以对流为主的方程特别有效。传统上，算法本身及其分析都与不同类型的正交函数的展开有关。最近出版的《伪谱方法实用指南》一书(作者)指出，通过将PS方法视为FD方法的特例，可以获得大量的推广、增强和见解。在赠款DMS-9706916项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，但径向基函数的光谱精度，加上其极大的几何灵活性，似乎使其非常适合于在不规则边界附近补充PS方法。要实现这一点，需要对径向基函数近似的基本特征进行大量研究，并研究它们与PS和FD方法的接口问题。伪谱(PS)方法最早是在20世纪70年代初由S提出的(与气象学和湍流模拟有关)。自那以后，它们已被证明对许多其他领域的高精度计算非常有效，例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的雷达散射、计算机芯片上组件之间的电干扰以及信号在光纤中的传输。PS方法的主要缺点是难以在复杂的几何问题中应用。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区域分割成更简单的部分，然后应用一些适当的计算方法来耦合子域)。在这项拨款下，将引入一种截然不同的、非常有前途的方法，然后进行探索。这涉及到将“经典”PS方法与几何上极其灵活的径向基函数(RBF)方法相结合。这项研究从未尝试过，而且属于“高风险、高收益”的范畴。成功绝非板上钉钉，但如果成功，可能会产生重大影响，特别是在计算电磁学方面，例如目标的雷达散射/隐身特性的模拟。

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.