A Fourier Approach to the Analysis and Design of Numerical Methods for Differencial Equations

常微分方程和偏微分方程的数值解提供了一类重要的大规模科学计算问题。傅里叶分析作为一种简单有效的工具被应用数学家用来分析偏微分方程和偏微分方程的近似格式和求解方法的数值性质。这包括三个主要目标：(1)利用用于数字滤波和数字控制领域的傅立叶工具来求解ODE和椭圆偏微分方程；(2)利用傅立叶技术分析数值ODE和椭圆偏微分方程算法，以便电气工程师在更熟悉的环境中理解它们；(3)开发新的并行算法，研究其在并行算法上的性能，研究其在连接机CM/2和Alliant FX/80等并行计算机上的性能。

The numerical solution of ordinary and partial differential equations (ODEs and PDEs) provides an important class of large scale scientific computing problems. Fourier analysis has been used by applied mathematicians as a simple and effective tool to analyze the numerical properties of approximation schemes and solution methods for ODEs and PDEs. This includes three main objectives: (1) exploiting Fourier tools used in the areas of digital filtering and digital control for solving ODEs and elliptic PDEs, (2) analyzing numerical ODE and elliptic PDE algorithms with Fourier techniques so that they can be understood by electrical engineers in a more familiar setting, and (3) development of new parallel algorithms and study of their performnce on parallel algorithms and study of their performance on parallel computers such as the Connection Machine CM/2 and Alliant FX/80.