Cahn-Hilliard方程在圓盤上的數值模擬

本論文想要介紹計算Cahn-Hilliard 方程在單位圓盤上的數值方法，因此我們必須要先解釋，如何解泊松方程, 雙調和方程以及四階擴散方程。我們首先利用傅立葉級數推導出常微分方程，並且利用二階有限差分法來解此方程，接下來我們離散它利用縮半格網格點的方法，這樣做是為了避開在原點所需的邊界條件。在柏松方程，我們利用緊緻四階有限差分法來解；在雙調和方程，我們利用三種不同的方法來解它，第一個方法是將雙調和方程拆成兩個柏松方程，第二個方法是直接將雙調和方程離散，第三個方法是利用緊緻四階有限差分法；在四階擴散方程，我們利用克蘭克－尼科爾森方法以及二階有限差分法離散。最後Cahn-Hilliard 方程的部分是利用反向尤拉法去解在圓盤上的Cahn-Hilliard 方程。

關鍵字

Cahn-Hilliard ；泊松方程；雙調和方程；四階擴散方程；有限差分法；克蘭克－尼科爾森方法；反向尤拉法

並列摘要

We want to solve a numerical method for the Cahn-Hilliard equation on a unit disk, so we will explain firstly how we solve Δu = f, Δ2u = f and ut + Δ2u = f. We first use the truncated Fourier series expansion to derive ODEs, then we solve those equations by second-order finite difference discretizations. Then, we discretize the resulting equations on a radial grid by shifting half mesh width in order to avoid the coordinate singularity at origin and endpoint. In the Poisson equation, we will solver it by the compact fourth-order finite difference scheme. In the biharmonic equation, we will try three solver. One is split biharmonic equation into two Poisson equations, another is discretization biharmonic equation directly, and the other is discretization it by the compact fourth-order finite difference scheme. In the Fourth order diffusion equation, we use the Crank-Nicolson method to discrete the equation. The part of the biharmonic equation, we chose to discrete directly. Finally, we use the backward Euler method to discretize the Cahn-Hilliard equation on a disk.