Numerical solution of PDEs by the spectral method

物理科学和工程中的许多问题(例如天气预报和飞机上空的气流)都被假设为偏微分方程(PDE)。但主要的挑战是准确和高效地求解这些偏微分方程组。谱方法是一类求解偏微分方程组的算法，它们被证明对于矩形区域是准确和高效的。然而，许多问题涉及不规则的区域。我们打算研究应用于不规则区域问题的谱方法。例如，作为一个具体的例子，我们研究高尔夫球上方的空气流动。例如，球上的凹陷给任何数值算法带来很大的困难。这个项目的第二个目标是创建一个通用软件，使用谱方法求解基本上任意几何上的线性偏微分方程组。这将为科学和工程界提供一个简单易用的程序，可以有效地求解他们的偏微分方程组。

Many problems in the physical sciences and engineering (weather forecasting and flow of air over an aircraft to name two examples) are posed as partial differential equations (PDEs).  The main challenge is to solve these PDEs accurately and efficiently.  Spectral method is a class of algorithms for solving PDEs and they have proven to be accurate and efficient for rectangular domains. However, many problems involve irregular domains.  We intend to study spectral methods applied to problems with irregular domains.  As a specific example, we study the flow of air over a golf ball.  The dimples on the ball cause great difficulties for any numerical algorithm. A second goal of this project is to create a general purpose software which solves linear PDEs on essentially arbitrary geometry using spectral methods.  This will provide the scientific and engineering community with an easy-to-use program which can solve their PDEs efficiently.