Numerical solution of PDEs by the spectral method

物理科学和工程中的许多问题（例如天气预报和飞行器上的气流）都是由偏微分方程（PDE）提出的。 主要的挑战是准确有效地求解这些偏微分方程。 谱方法是求解偏微分方程的一类算法，在矩形区域上具有很高的精度和效率。然而，许多问题涉及不规则域。 我们打算研究谱方法应用于不规则区域的问题。 作为一个具体的例子，我们研究了高尔夫球上的空气流动。 球上的凹坑给任何数值算法带来了很大的困难。这个项目的第二个目标是创建一个通用的软件，解决线性偏微分方程的基本上任意的几何形状使用谱方法。 这将为科学和工程界提供一个易于使用的程序，可以有效地解决他们的偏微分方程。

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.