內嵌介面法求解有不連續條件的三維Poisson方程式

Abstract

在本文中，我們將一個簡單的內嵌介面法如何求解有不連續條件的二維Poisson方程式推廣到三維。此數值計算方法主要基於沿法線方向運用泰勒展開式，並將解的不連續條件嵌入有限差分項中。之後，我們就可以使用一些有效率的疊代法來求解此離散線性系統，例如：共軛梯度法。由數值計算結果可看出此方法有二階精度。

In this paper, we extend a simple version of the immersed interface method (IIM) for 2D Poisson problems to 3D with jump conditions across the interface. The numerical method is based on applying the Taylor's expansions along the normal direction and incorporating the solution and its normal derivative jumps into the finite difference approximations. Then, we can apply some efficient iterative solvers such as the conjugate gradient method to solve the discretized Laplacian linear system. The numerical results show that the scheme is indeed second-order accurate.