极坐标系下泊松方程的拟谱方法

A new pseudo-spectral method for solving Poisson equation in polar coordinate system

极坐标系下的泊松方程,由于坐标原点的奇异性,给谱方法的实施带来了很大的困难。本文提出了一种新的拟谱方法,在径向上求解区域为[-1,1],采用标准的Gauss-Lobatto配置点;而在角方向上配置点均匀分布在[0,π]内。通过对配置点及空间导数矩阵的处理,成功解决了坐标奇异问题。同时也避免了配置点在原点附近的集中,极大改善了矩阵条件数,减小了舍入误差,从而提高了解的精度。数值实验表明,该方法具有很高的精度。

Due to the existence of the coordinate singularity, there is some difficulty to implement spectral method in polar coordinate. A new pseudo-spectral method for solving Poisson equation in polar coordinate system is presented. In radical direction, Gauss-Lobatto collocation points are adopted in [ - 1,1 ]; in peripheral direction, all the points are uniformly distributed in [0,π]. After carefully arranging the collocation points and the matrix of derivative, the coordinate singularity is circumvented. Also, the clustering of collocation points near the center is avoided. Hence, the condition number is strongly improved and the round-off error is reduced. The numerical experiments demonstrate that the new pseudo-spectral method has a high accuracy.