JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (6): 118-126.doi: 10.6040/j.issn.1671-9352.0.2018.146

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Error analysis of a linearized Crank-Nicolson scheme for the phase field crystal equation

LI Juan   

  1. College of Jinshen, Nanjing Audit University, Nanjing 210023, Jiangsu, China
  • Published:2019-06-05

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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 ux)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

CLC Number: 

  • O241.82
[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.
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