三維Poisson方程在圓柱及球座標下之形式四階緊緻差分法

Abstract

在這篇論文裡，將會介紹三維 Poisson 方程在圓柱及球座標下簡單且有效率的四階緊緻解法。這個解法是由截斷(truncated)傅利葉級數展開式所產生，且得到一組傅利葉係數所形成的偏微分方程組，運用緊緻差分技巧，我們可以得到四階精確且不需奇異點條件的結果。接著利用兩種有效的迭代法(GMRES,BI-CGSTAB)來解離散後，傅利葉係數所形成非對稱的線性系統並配合不同的preconditioner。

A simple and efficient compact fourth-order Poisson solver in cylindrical and spherical coordinates is presented. The solver relies on the truncated Fourier series expansion, where the differential equations of Fourier coefficients have been solved by fourth-order finite difference discretizations without pole conditions. And two kinds of efficient iterative method, GMRES and Bi-CGSTAB, with different preconditioners are applied to solve the resulted nonsymmetrical systems of Fourier coefficients.