內嵌介面問題的數值方法與應用

Abstract

本論文介紹計算有不連續條件的二維Poisson方程的一個簡單快速的數值方法，並應用本方法來解Stokes方程和Navier-Stokes方程。在本文中，提供在計算區域上，處理介面的幾何形狀以及在介面上方程的不連續條件。我們使用三次雲線內插來近似不連續條件所在之介面，以及使用二階精度的外插公式來處理不連續條件。在效率的考量上，使用快速的Poisson算法。我們給出一些計算數據，來觀察此方法的精度以及應用。

In this paper, we introduce a simple and efficient second order numerical method to solve Poisson equation with jump conditions in two-dimension (2-D). We also apply the method to simulate Stokes and unsteady Navier-Stokes flow dynamics. One of the essential components of the method is to handle the geometric of the immersed interface and jump discontinuities along it in the computational domain. Cubic spline interpolation is implemented to deal with the interface and a second order accurate extrapolation is applied to incorporate these jump conditions. We also apply the marker-and-cell (MAC) method cooperating with the interface. In the matter of efficiency, we can apply some fast Poisson solvers such as Fishpack. We verify the accuracy, efficiency of our method by showing some numerical experiments.