Collocation spectral methods in the solution of poisson equation

• Main Author: Su, Yuhong
• Language: English
• Published: 2009
• Online Access: http://hdl.handle.net/2429/8275

The Poisson equation is a very important partial differential equation for many branches of science and engineering. A fast, robust and accurate Poisson equation solver can find immediate applications in many fields such as electrical engineering, plasma physics, incompressible fluid mechanics and space science. In this thesis, the collocation pseudospectral method is employed to solve Poisson equation. Many papers propose how to apply spectral methods to solve partial differential equations in Cartesian coordinates, but not much attention has been paid to the use of spectral methods in polar coordinates and cylindrical coordinates. In this thesis, the application of the collocation spectral methods to the solution of one and two-dimensional Poisson equations on a rectangular domain in Cartesian coordinates; and on a disk, a part of disk and an annulus in polar coordinates. Also considered is the three-dimensional Poisson equation in a cube in Cartesian coordinates; and a cylinder, a cylindrical annulus and part of a cylinder in cylindrical coordinates. Two of the most important approaches in this thesis are: First we put forward a new collocation spectral method that can avoid the coordinate singularities and solve the Poisson equation in a disk directly by the eigenvalue technique. This can simplify the use of the spectral method in polar coordinates and cylindrical coordinates, where coordinate singularities cause problems for spectral methods. Second, we also give an algorithm that can directly solve the discrete Poisson equation in cylindrical coordinates after discretization by the collocation spectral method. Basically, the idea is that we combine r and z directions together and transform the equation into a form that can be solved by an eigenvalue technique. This method is very fast and efficient.