极坐标与圆柱坐标下Fourier-Chebyshev配置点谱方法泊松方程求解器

采用矩阵相乘的Fourier-Chebyshev配置点谱方法求解极坐标与圆柱坐标系下的泊松方程.通常,在极坐标与圆柱坐标系下运用谱方法求解泊松方程会产生奇点问题.为了避免这个问题,分别采用两种方法开发了泊松方程求解器.一种方法是采用Gauss-Radau配置点,从而排除中心点r=0;另一种方法是采用区域转换将半径方向计算域0,1]转换成-1,1],采用Gauss-Lobatto配置点,当节点数取奇数时同样避开了中心点r=0.这两种方法均避免了中心处的奇点,且不需构造额外的极条件.针对二维、三维的不同算例进行了比较和验证计算.计算结果证明两个求解器都具有直接、快速、高精度的特性.

Fourier-Chebyshev Collocation Spectral Poisson Solvers in Polar and Cylindrical Coordinate Systems

The Poisson solvers in polar and cylindrical coordinate systems are developed using Fourier-Chebyshev collocation spectral method based on matrix-matrix multiplication.Usually the singularities will appear in the solution to Poisson equation in polar and cylindrical coordinate systems by spectral method.To avoid such a problem,two methods are proposed to solve the Poisson equation.The first is introducing the Gauss-Radau collocation points,thus excluding the singularity at the origin.The second method is transforming the computing interval in radial direction into and then introducing the Gauss-Lobatto collocation points,so as to exclude the singularity at the origin when the number of nodes is odd.And no extra pole conditions are required by both methods to avoid the singularity at the origin.The two methods are compared with each other and verified via 2D and 3D instances separately,and the results indicate that they are direct,fast,and highly accurate.