×

A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect. (English) Zbl 1436.74049

Summary: In this paper, we extend the phase field crystal model to the modified phase field crystal model which includes diffusive dynamics, elastic interaction, and stochastic noises effect. We present a second-order accurate semi-implicit finite difference scheme for the modified phase field crystal model. The resulting scheme is based on the stabilized splitting method and Crank-Nicolson method. The nonlinear term is linearized by the Taylor series. The resulting scheme is linear at each time step, which makes it easy to be implemented and efficient to be solved by using the linear multigrid solver. We prove that the resulting scheme is unconditionally energy stable. Various numerical experiments are conducted to verify the accuracy and efficiency of our proposed algorithm.

MSC:

74N05 Crystals in solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] de Gennes, P. G., Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72, 9, 4756-4763 (1980) · Zbl 1110.82310
[2] Shiwa, Y., The amplitude and phase-diffusion equations for lamellar patterns in block copolymers, Phys. Lett. A., 228, 279-282 (1997) · Zbl 1042.82639
[3] Kawasaki, K.; Ohta, T.; Kohrogui, M., Equilibrium morphology of block copolymer melts 2, Macromolecules, 19, 10, 2621-2632 (1986)
[4] Aristotelous, A. C.; Karakashian, O.; Wise, S. M., A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver, Discrete Contin. Dyn. Syst. Ser. B, 18, 9, 2211-2238 (2013) · Zbl 1278.65149
[5] Gomez, H.; Hughes, T. J.R., Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models, J. Comput. Phys., 230, 13, 5310-5327 (2011) · Zbl 1419.76439
[6] Gomez, H.; Nogueira, X., An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249-252, 52-61 (2012) · Zbl 1348.74280
[7] Liu, C.; Shen, J.; Yang, X. F., Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density, J. Sci. Comput., 62, 601-622 (2015) · Zbl 1326.76064
[8] Cheng, Q.; Yang, X. F.; Shen, J., Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model, J. Comput. Phys., 341, 44-60 (2017) · Zbl 1380.65203
[9] Jeong, D.; Shin, J.; Li, Y. B.; Choi, Y.; Jung, J. H.; Lee, S., Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers, Curr. Appl. Phys., 14, 1263-1272 (2014)
[10] Choksi, R.; Peletier, M. A.; Williams, J. F., On the phase diagram for microphase separation of diblock copolymers-an approach via a nonlocal Cahn-Hilliard function, SIAM J. Appl. Math., 69, 6, 1712-1738 (2009) · Zbl 1400.74089
[11] Elder, K. R.; Katakowski, M.; Haataja, M.; Grant, M., Modeling elasticity in crystal growth, Phys. Rev. Lett., 88, 24, Article 245701 pp. (2002)
[12] Elder, K. R.; Grant, M., Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70, 5, Article 051605 pp. (2004)
[13] Zhang, Z.; Ma, Y.; Qiao, Z., An adaptive time-stepping strategy for solving the phase field crystal model, J. Comput. Phys., 249, 204-215 (2013) · Zbl 1305.82009
[14] Vignal, P.; Dalcin, L.; Brown, D. L.; Collier, N.; Calo, V. M., An energy-stable convex splitting for the phase-field crystal equation, Comput. Struct., 158, 355-368 (2015)
[15] Wise, S. M.; Wang, C.; Lowengrub, J. S., An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47, 3, 2269-2288 (2009) · Zbl 1201.35027
[16] Hu, Z.; Wise, S. M.; Wang, C.; Lowengrub, J. S., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, J. Comput. Phys., 228, 15, 5323-5339 (2009) · Zbl 1171.82015
[17] Yang, X. F.; Han, D. Z., Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model, J. Comput. Phys., 330, 1116-1134 (2017) · Zbl 1380.65209
[18] Li, Y. B.; Kim, J., An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 319, 194-216 (2017) · Zbl 1439.76122
[19] Li, Y. B.; Luo, C. J.; Xia, B. H.; Kim, J., An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces, Appl. Math. Model., 67, 477-490 (2019) · Zbl 1481.82014
[20] Yang, X. F.; Zhao, J., Efficient linear schemes for the nonlocal Cahn-Hilliard equation of phase field models, Comput. Phys. Comm., 235, 234-245 (2019) · Zbl 07682903
[21] Zhao, J.; Yang, X.; Gong, Y.; Wang, Q., A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg., 318, 803-825 (2017) · Zbl 1439.76124
[22] Gong, Y.; Zhao, J.; Wang, Q., Second order fully discrete energy stable methods on staggered grids for hydrodynamic phase field models of binary viscous fluids, SIAM J. Sci. Comput., 40, 2, 528-553 (2018) · Zbl 1393.65012
[23] Bueno, J.; Starodumov, I.; Gomez, H.; Galenko, P.; Alexandrov, D., Three dimensional structures predicted by the modified phase field crystal equation, Comput. Mater. Sci., 111, 310-312 (2016)
[24] Stefanovic, P.; Haataja, M.; Provatas, N., Phase field crystal study of deformation and plasticity in nanocrystalline materials, Phys. Rev. E, 80, 4, Article 046107 pp. (2009)
[25] Stefanovic, P.; Haataja, M.; Provatas, N., Phase-field crystals with elastic interactions, Phys. Rev. Lett., 96, 22, Article 225504 pp. (2006)
[26] Baskaran, A.; Hu, Z. Z.; Lowenfrub, J. S.; Wang, C.; Wise, S. M.; Zhou, P., Energy stabel and efficient finite-difference nonlinear multigrid schemesa for the modified phased crystal equation, J. Comput. Phys., 250, 270-292 (2013) · Zbl 1349.65265
[27] Galenko, P. K.; Gomez, H.; Kropotin, N. V.; Elder, K. R., Unconditionally stable method and numerical solution of the hyperbolic phase-field crystal equation, Phys. Rev. E, 88, Article 013310 pp. (2013)
[28] 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, Comput. Methods Appl. Mech. Engrg., 298, 453-484 (2016) · Zbl 1423.76321
[29] Lee, H. G.; Shin, J.; Lee, J. Y., First- and second-order energy stable methods for the modified phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 321, 1-17 (2017) · Zbl 1439.74240
[30] Li, T. J.; Zhang, P. W.; Zhang, W., Nucleation rate calculation for the phase transition of diblock copolymers under stochastic Cahn-Hilliard dynamic, SIAM Multi. Model. Simul., 11, 1, 385-409 (2013) · Zbl 1269.82073
[31] Zhang, W.; Li, T. J.; Zhang, P. W., Numerical study for the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics, Commun. Math. Sci., 10, 4, 1105-1132 (2012) · Zbl 1284.60139
[32] Han, D. Z.; Brylev, A.; Yang, X. F.; Tan, Z. J., Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two-phase incompressible flows, J. Sci. Comput., 70, 965-989 (2017) · Zbl 1397.76070
[33] Yang, X. F.; Zhao, J.; He, X. M., Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method, J. Comput. Appl. Math., 343, 80-97 (2018) · Zbl 1462.65117
[34] Li, Y. B.; Choi, Y.; Kim, J., Computationally efficient adaptive time step method for the Cahn-Hilliard equation, Comput. Math. Appl., 73, 1855-1864 (2017) · Zbl 1372.65235
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.