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浙江大学学报（理学版）
Journal of Zhejiang University (Science Edition)  2022, Vol. 49 Issue (1): 60-65    DOI: 10.3785/j.issn.1008-9497.2022.01.009
Mathematics and Computer Science     
A second-order linearized finite difference method for a higher order convective Cahn-Hilliard type equation
Juan LI()
Department of Basis Course, Nanjing Audit University Jinshen College, Nanjing 210023, China
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Abstract  

The higher order convective Cahn-Hilliard type equation is a kind of sixth-order evolution equation with fourth-order nonlinearity term.A linearized finite difference scheme is presented by using Taylor formula.The first time level is a two-layers implicit scheme,while the other time levels are three-layers implicit schemes.In the derivation of the scheme,nonlinear terms are discretized by central difference quotient.A theoretical analysis is carried out by the energy argument and mathematical induction. The uniqueness and convergence of the numerical solution are proved in L2 norm rigorously.The convergence order is two in time and space.Some numerical results are presented to demonstrate the efficiency of the difference scheme.

Key words： higher order Cahn-Hilliard type equation      linearized difference scheme      uniqueness      convergence      nonlinear problem      linearization     
Received: 01 July 2020      Published: 18 January 2022
CLC:  O 241.82  
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Juan LI
Cite this article:

Juan LI. A second-order linearized finite difference method for a higher order convective Cahn-Hilliard type equation. Journal of Zhejiang University (Science Edition), 2022, 49(1): 60-65.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I1/60


高阶对流Cahn-Hilliard型方程的二阶线性化差分方法

高阶对流Cahn-Hilliard型方程是一类空间六阶且具有四阶非线性项的发展方程。首先，给出了线性化差分格式，其第一时间层为2层隐式差分格式，其余时间层为3层隐式差分格式。其次，在差分格式建立过程中，利用中心差商对四阶非线性项进行离散，证明了差分格式解的唯一性和收敛性，并得到其在时间和空间上的收敛阶均为二阶。最后，通过数值算例，验证了差分格式的有效性。


关键词： 高阶对流Cahn-Hilliard型方程,  线性化差分格式,  唯一性,  收敛性,  非线性问题,  线性化 
NH2（h，τ）order1H∞（h，τ）order2
202.451×10-41.8711.510×10-41.859
406.703×10-51.8784.162×10-51.872

80

160

1.824×10-5

4.949×10-6

1.882

/

1.137×10-5

3.113×10-6

1.869

/
Table 1 The errors and temporal convergence orders of the difference scheme in L2 -norm and L∞-norm when T=5， M=2 000，ε=0.5
MH2（h，τ）order3H∞（h，τ）order4
109.914×10-21.8412.245×10-21.855
202.767×10-21.9506.208×10-31.808

40

80

7.160×10-3

1.825×10-3

1.972

/

1.773×10-3

4.559×10-4

1.959

/
Table 2 The errors and spatial convergence orders of the difference scheme in L2 -norm and L∞-norm when T=5， N=2 000，ε=0.5
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