【摘要】：Cahn-Hilliard方程是四階非線性擴(kuò)散方程,本文主要討論該方程的有限元離散格式以及數(shù)值解,分別對(duì)一維情形和二維情形采用了連續(xù)元和局部間斷有限元兩種方法來求解,對(duì)一維情形,采用先對(duì)空間做連續(xù)有限元和局部間斷有限元兩種逼近,得出兩種半離散格式,并引進(jìn)能量函數(shù)證明了格式的穩(wěn)定性,再對(duì)時(shí)間用向前歐拉差分得出全離散格式,對(duì)二維情形,同樣對(duì)空間做了連續(xù)有限元和局部間斷有限元兩種逼近,再對(duì)時(shí)間分別用了 Crank-Nicolson離散法,三階TVD Runge-Kutta離散方法,得出全離散格式。文章最后一部分,對(duì)一維情形兩種格式做了數(shù)值解的計(jì)算,計(jì)算了給定的非線性項(xiàng)和邊界條件下,不同初值情形下的數(shù)值解,驗(yàn)證了兩種數(shù)值格式能夠保證質(zhì)量守恒以及能量衰減,同時(shí)計(jì)算過程中也發(fā)現(xiàn)局部間斷有限元的顯格式在解我們給定的方程時(shí)不如連續(xù)元穩(wěn)定,對(duì)時(shí)間步長(zhǎng)要求更加嚴(yán)格,容易爆破。
[Abstract]:The Cahn-Hilliard equation is a fourth-order nonlinear diffusion equation. In this paper, the finite element discrete scheme and numerical solution of the equation are mainly discussed. The continuous element method and local discontinuous finite element method are used to solve the one-dimensional and two-dimensional cases, respectively. Two semi-discrete schemes are obtained by means of continuous finite element method and local discontinuous finite element approximation to space, and the stability of the scheme is proved by introducing energy function, and then the full discrete scheme is obtained by forward Euler difference for time, and the two-dimensional case is obtained. The space is approximated by continuous finite element method and local discontinuous finite element method respectively. Then the Crank-Nicolson discrete method and the third order TVD Runge-Kutta discretization method are used to obtain the full discrete scheme for time. In the last part of the paper, the numerical solutions of two schemes in one-dimensional case are calculated, and the numerical solutions under the given nonlinear terms and boundary conditions are calculated under different initial conditions. It is proved that the two numerical schemes can guarantee the conservation of mass and energy attenuation. It is also found that the explicit scheme of local discontinuous finite element is less stable than the continuous element in solving the equation given by us, and the time step is more strict. It is easy to blow up.
【學(xué)位授予單位】：湘潭大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.82

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相關(guān)期刊論文 前2條

1 葉興德,程曉良;Cahn—Hilliard方程的Legendre譜逼近[J];計(jì)算數(shù)學(xué);2003年02期

2 ;AN EXPLICIT PSEUDO-SPECTRAL SCHEME WHIT ALMOST UNCONDITIONAL STABILITY FOR THE CAHN-HILLIARD EQUATION[J];Journal of Computational Mathematics;2000年02期

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本文編號(hào)：2184283