晶体相场方程的线性化Crank-Nicolson格式的误差分析

doi:10.6040/j.issn.1671-9352.0.2018.146

摘要/Abstract

摘要： 晶体相场模型是一类空间六阶非线性发展方程。首先,给出了线性化Crank-Nicolson格式,该格式在第一、二时间层是显式差分格式,其余时间层是线性化隐式差分格式。在建立差分格式的过程中,将非线性项(u3)_xx改写成(3u2 u_x)_x,利用中心差商对其进行离散。其次,证明了差分格式解的先验估计式及无条件收敛性,收敛阶在时空方向均为二阶。最后通过数值算例,验证差分格式是有效的。

关键词: 晶体相场方程, 线性化Crank-Nicolson 格式, 收敛性, 非线性问题, 线性化

Abstract: The phase field crystal model is a high order nonlinear evolutionary equation with the sixth order derivative in space. A linearized Crank-Nicolson scheme is presented. The scheme is explicit at the first-and second-time level. We just only to solve an implicit linearized scheme at the rest of the time level. In the derivation of the scheme, the nonlinear term (u3)_xx is rewritten to be (3u2 u_x)_x, and then be discretized by central difference quotient. The priori estimate of the numerical solution and unconditional convergence is proved in L2 norm. The convergence order is two in time and space. Some numerical examples are presented to demonstrate the theoretical results.

Key words: phase field crystal equation, linearized Crank-Nicolson scheme, convergence, nonlinear problem, linearization

中图分类号:

O241.82

李娟. 晶体相场方程的线性化Crank-Nicolson格式的误差分析[J]. 《山东大学学报(理学版)》, 2019, 54(6): 118-126.

LI Juan. Error analysis of a linearized Crank-Nicolson scheme for the phase field crystal equation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 118-126.

参考文献

[1] ELDER K R, KATAKOWSKI M, HAATAJA M, et al. Modeling elasticity in crystal growth[J]. Phys Rev Lett, 2002, 88(24):245701.
[2] ELDER K R, GRANT M. Modeling elastic and plastic deformations in nonequilbrium processing using phase field crystals[J]. Phys Rev E, 2004, 70(1):051605.
[3] WISE S M, WANG Cheng, LOWENGRUB J S. An energy-stable and convergent finite-difference scheme for the phase field crystal equation[J]. SIAM J Numer Anal, 2009,47(3):2269-2288.
[4] ] HU Zhengzheng, WISE S M, WANG Cheng, et al. Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation[J]. J Comput Phys, 2009, 228(15):5323-5339.
[5] GOMEZ H, NOGUEIRA X. An unconditionally energy-stable method for the phase field crystal equation[J]. Comput Methods Appl Mech Engrg, 2012, 249/250/251/252:52-61.
[6] ZHANG Zhengru, MA Yuan, QIAO Zhonghua. An adaptive time-stepping strategy for solving the phase field crystal model[J]. J Comput Phys, 2013, 249:204-215.
[7] SHIN J, LEE H G, LEE J Y. First and second order numerical methods based on a new convex splitting for phase-field crystal equation[J]. J Comput Phys, 2016, 327:519-542.
[8] CAO Haiyan, SUN Zhizhong. Two finite difference schemes for the phase field crystal equation[J]. Sci China Math, 2015, 58(11):2435-2454.
[9] YANG Xiaofeng, HAN Daozhi. Linearly first-and second-order, unconditionally energy stable schemes for the phase field crystal model[J]. J Comput Phys, 2017, 330:1116-1134.
[10] 李娟. 粘性Cahn-Hilliard方程的半线性Crank-Nicolson格式[J]. 四川师范大学学报(自然科学版), 2018, 41(2):237-245. LI Juan. On a semi-linearized Crank-Nicolson scheme for the viscous Cahn-Hilliard Equation[J]. Journal of Sichuan Normal University(Natural Science), 2018, 41(2):237-245.
[11] 孙志忠. 偏微分方程数值解法[M]. 2版. 北京: 科学出版社, 2012: 13-16. SUN Zhizhong. The method to numerical solutions of partial differential equations[M]. 2nd ed. Beijing: Science Press, 2012: 13-16.
[12] LI Juan, SUN Zhizhong, ZHAO Xuan. A three level linearized compact difference scheme for the Cahn-Hilliard equation[J]. Sci China Math, 2012, 55(4):805-826.
[13] STEFANOVIC P, HAATAJA M, PROVATAS N. Phase-field crystals with elastic interactions[J]. Phys Rev Lett, 2006, 96(22):1-4.
[14] STEFANOVIC P, HAATAJA M, PROVATAS N. Phase field crystal study of deformation and plasticity in nanocrystalline materials[J]. Phys Rev E, 2009, 80:046107.
[15] WANG Cheng, WISE S M. An energy stable and convergent finite-difference scheme for the modified phase field crystal equation[J]. SIAM J Numer Anal, 2011, 49(3):945-969.
[16] BASKARAN A, HU Zhengzheng, LOWENGRUB J S, et al. Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation[J]. J Comput Phys, 2013, 250:270-292.
[17] GALENKO P K, GOMEZ H, KROPOTIN N V, et al. Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation[J]. Phys Rev E, 2013, 88:013310.
[18] BASKARAN A, LOWENGRUB J S, WANG Cheng, et al. Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation[J]. SIAM J Numer Anal, 2013, 51(5):2851-2873.
[19] DEHGHAN M, MOHAMMADI V. The numerical simulation of the phase field crystal(PFC)and modified phase field crystal(MPFC)models via global and local meshless methods[J]. Comput Methods Appl Mech Engrg, 2016, 298:453-484.
[20] LEE H G, SHIN J, LEE J Y. First-and second-order energy stable methods for the modified phase field crystal equation[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 321:1-17.

相关文章 15

[1]	李素丽,谢华朝. 非线性Sobolev-Galpern型方程的超收敛分析[J]. 《山东大学学报(理学版)》, 2026, 61(4): 133-142.
[2]	周小英,吉晨,涂晓艺. 基于线性化技术的变点分位数回归模型的估计与应用[J]. 《山东大学学报(理学版)》, 2025, 60(3): 69-76.
[3]	李学文,冯可馨,王小刚. 删失分位数回归模型中的多变点估计[J]. 《山东大学学报(理学版)》, 2025, 60(2): 96-104.
[4]	温淑鸿,柯艺芬,黎科良. 求解隐式互补问题的模系矩阵分裂迭代方法[J]. 《山东大学学报(理学版)》, 2025, 60(12): 55-65.
[5]	王洋. 一类复对称线性方程组的块三角分裂及其预处理迭代算法[J]. 《山东大学学报(理学版)》, 2024, 59(10): 1-9.
[6]	嘉程程,吴群英. 次线性期望空间下END序列加权和的完全收敛性[J]. 《山东大学学报(理学版)》, 2022, 57(10): 79-87.
[7]	王松华,罗丹,黎勇. 一类新型的修正WYL共轭梯度算法[J]. 《山东大学学报(理学版)》, 2021, 56(9): 87-95.
[8]	张如,韩旭,刘小刚. 非线性延迟微分方程边值方法的收敛性和收缩性[J]. 《山东大学学报(理学版)》, 2019, 54(8): 97-101.
[9]	张泰年,李照兴. 一类退化抛物型方程反问题的收敛性分析[J]. 山东大学学报（理学版）, 2017, 52(8): 35-42.
[10]	郑秀云,史加荣. Armijo型线搜索下的全局收敛共轭梯度法[J]. 山东大学学报（理学版）, 2017, 52(1): 98-101.
[11]	王开荣,高佩婷. 建立在DY法上的两类混合共轭梯度法[J]. 山东大学学报（理学版）, 2016, 51(6): 16-23.
[12]	张玉,肖犇琼,许可,沈爱婷. NSD随机变量阵列的完全矩收敛性[J]. 山东大学学报（理学版）, 2016, 51(6): 30-36.
[13]	张立君,郭明乐. 行为渐近负相协随机变量阵列加权和的矩完全收敛性[J]. 山东大学学报（理学版）, 2016, 51(2): 42-49.
[14]	徐言超. 连续时间正系统的静态输出反馈鲁棒H_∞控制[J]. 山东大学学报（理学版）, 2016, 51(12): 87-94.
[15]	谭闯, 郭明乐, 祝东进. 行为ND随机变量阵列加权和的矩完全收敛性[J]. 山东大学学报（理学版）, 2015, 50(06): 27-32.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed