Poisson 方程在不規則區域的高階緊緻差分法

Abstract

在這篇論文中，我們考慮在不規則區域上有著Dirichlet邊界條件的二維Poisson 方程，而此不規則區域將被限制在一個矩形區域中來做計算。在原不規則區域內的網格點，我們的高階精度方法是用標準的緊緻九點格式來離散此Poisson方程，但對於靠近邊界的點將要做些特別處理。在這些需要另外做處理的點，我們利用外插法來造出仿真的值。使用的外插法有常數、線性以及二次的外插法，並將分別可以得到一階、二階以及三階的精度。其中常數及線性的外插法，可以使要解的線性系統仍保持對稱性。

In this thesis, we consider the 2D Poisson equation subject to Dirichlet boundary conditions on an irregular domain. The region of interest is embedded in a rectangular domain. For our higher-order accurate scheme, at internal grid points, the Poisson equation is discretized with the standard compact nine point stencil with special treatment at the edges. At the irregular point, we define ghost value constructed by extrapolations. This yields first, second and third order accuracy in the case of the constant, linear and quadratic extrapolations, respectively. In the case of constant and linear extrapolations, the linear system is symmetric.