Allen-Cahn與Cahn-Hilliard方程的數值研究

Abstract

本論文介紹計算Allen-Cahn方程的一個簡單快速且穩定的有限差分算法，只需要解一個三條對角線的線性方程，並利用本方法做調整來解Cahn-Hilliard方程。我們也介紹一套利用快速傅立葉轉換的優點來計算Allen-Cahn方程的分離算法。我們修正了Allen-Cahn方程原本質量不守衡的問題，並且將這套方法應用在不可壓縮的兩項流混合問題上。

We propose simple finite difference schemes with fast and stable computations for the Allen-Cahn equation with solving tri-diagonal matrices. We also deal this with the Cahn-Hilliard equation in a similar way. We introduce a special splitting method for the Allen-Cahn equation in advantage of applying fast Fourier transform. Mass-conservation modification for the Allen-Cahn equation is provided. And we apply these methods to compute the mixture problem of two incompressible fluids.