Cahn-Hilliard方程的有限元离散方法及数值解研究
发布时间：2018-08-15 13:02
【摘要】：Cahn-Hilliard方程是四阶非线性扩散方程,本文主要讨论该方程的有限元离散格式以及数值解,分别对一维情形和二维情形采用了连续元和局部间断有限元两种方法来求解,对一维情形,采用先对空间做连续有限元和局部间断有限元两种逼近,得出两种半离散格式,并引进能量函数证明了格式的稳定性,再对时间用向前欧拉差分得出全离散格式,对二维情形,同样对空间做了连续有限元和局部间断有限元两种逼近,再对时间分别用了 Crank-Nicolson离散法,三阶TVD Runge-Kutta离散方法,得出全离散格式。文章最后一部分,对一维情形两种格式做了数值解的计算,计算了给定的非线性项和边界条件下,不同初值情形下的数值解,验证了两种数值格式能够保证质量守恒以及能量衰减,同时计算过程中也发现局部间断有限元的显格式在解我们给定的方程时不如连续元稳定,对时间步长要求更加严格,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.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
本文编号：2184283
[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.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82
【参考文献】
相关期刊论文 前2条
1 叶兴德,程晓良;Cahn—Hilliard方程的Legendre谱逼近[J];计算数学;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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