李嫻娟
(福州大學數學與計算機科學學院,福建 福州 350116)
?
有限差分-譜方法求解Allen-Cahn方程的誤差分析
李嫻娟
(福州大學數學與計算機科學學院,福建 福州 350116)
Allen-Cahn方程; 有限差分; 譜方法; 誤差分析
考慮Allen-Cahn方程:
其中:Ω?d,d=2, 3為有界區域;u表示合金的密度;ε表示交面厚度;n為外法向;f(u)=f′(u),F(u)為能量勢. 齊次Neumann邊界條件說明合金在邊界墻外沒有質量損失. Allen-Cahn方程可視為Liapunov能量泛函E(u)在L2中的梯度流.
(f(a)-f(b), a-b)≥γ1(f ′(a)(a-b), a-b)-
(ΠNu-u,vN)=0 (?vN∈PN)
(
引理1 設Allen-Cahn算子LAC:=-Δ+f′(u)I的特征值滿足:
有特征值估計
根據式(7)、 (10)、 (12)以及式(14)得:
綜合式(11)以及式(15)可得特征值估計式(13)成立.
則
其中:δt以及N滿足
證明 將方程(1)兩端同時乘以vN并在Ω上積分得:
(ut,vN)+(u,
對方程(20)取t=tn+1,并與方程(16)相減得:
接下來逐項估計式(22).
由Cauchy-Schwarz不等式以及Young不等式得:
由引理1以及假設(5)得:
將式(23)~(27)代入方程(22)得:
對方程(28)兩端同時求和并利用不等式(8)~(9), 文獻[6]中性質2以及式(47)得:
注意到(19a), 則有
最后根據歸納法證明結論, 假設:
根據方程(30)并應用Gronwall不等式得:
[1] ANDERSON D M, MCFADDEN G B, WHEELER A A. Diffuse-interface methods in fluid mechanics[J]. Annual Review of Fluid Mechanics, 1998, 30(1): 139-165.
[2] DU Q, LIU C, WANG X Q. Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions[J]. Journal of Computational Physics, 2006, 212(2): 757-777.
[3] LIU C, SHEN J. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method[J]. Physica D, 2003, 179(3/4): 211-228.
[4] SHEN J, YANG X F. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[J]. Discrete and Continuous Dynamical Systems, 2010, 28(4): 1 669-1 691.
[5] DU Q, NICOLAIDES R A. Numerical analysis of a continuum model of phase transition[J]. SIAM Journal on Numerical Analysis, 1991, 28(5): 1 310-1 322.
[6] FENG X B, PROHL A. Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows[J]. Numerische Mathematik, 2003, 94(1): 33-65.
[7] YE X D. The legendre collocation method for the Cahn-Hilliard equation[J]. Journal of Computational and Applied Mathematics, 2003, 150(1): 87-108.
[8] CANUTO C, HUSSAINI M Y, QUARTERONI A,etal. Spectral methods: fundamentals in single domains[M]. Berlin: Springer, 2006.
(責任編輯: 林曉)
Error analysis of finite difference -spectral method for Allen-Cahn equations
LI Xianjuan
(College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116,China)
We investigate the error analysis of finite difference/spectral method for Allen-Cahn equations, where a vary small parameterεrepresents the interfacial width. The error analysis shows that if the time stepδtis small enough and polynomial orderNis large enough, then the error bound of the fully discrete schemes depends on the termε-σwhen the regularity of initial datau0depends on the same termε-σ. This result improves the error bound in which depends on e
Allen-Cahn equation; finite difference method; spectral method; error analysis
10.7631/issn.1000-2243.2017.03.0307
1000-2243(2017)03-0307-05
2016-11-05
李嫻娟(1982-),副教授,主要從事偏微分方程數值解方面研究,xjli@fzu.edu.cn
福建省教育廳科技資助項目(JA14034); 福建省自然科學基金資助項目(2016J01013)
O241.8; O241.1
A