×

Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation. (English) Zbl 1409.82014

Summary: In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in Hu et al. (2009), with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
82D25 Statistical mechanics of crystals
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Elder, K.; Katakowski, M.; Haataja, M.; Grant, M., Modeling elasticity in crystal growth, Phys. Rev. Lett., 88, 245701 (2002)
[2] Elder, K.; Grant, M., Modeling elastic and plastic deformations in nonequilibrium processing using phase filed crystal, Phys. Rev. E, 90, 051605 (2004)
[3] Stefanovic, P.; Haataja, M.; Provatas, N., Phase-field crystals with elastic interactions, Phys. Rev. Lett., 96, 225504 (2006)
[4] Elder, K.; Provatas, N.; Berry, J.; Stefanovic, P.; Grant, M., Phase-field crystal modeling and classical density functional theory of freezing, Phys. Rev. B, 77, 064107 (2007)
[5] Provatas, N.; Dantzig, J.; Athreya, B.; Chan, P.; Stefanovic, P.; Goldenfeld, N.; Elder, K., Using the phase-field crystal method in the multiscale modeling of microstructure evolution, JOM, 59, 83 (2007)
[6] Backofen, R.; Rätz, A.; Voigt, A., Nucleation and growth by a phase field crystal (PFC) model, Phil. Mag. Lett., 87, 813 (2007)
[7] Marconi, U.; Tarazona, P., Dynamic density functional theory of liquids, J. Chem. Phys., 110, 8032 (1999)
[8] Elder, K.; Provatas, N., Amplitude expansion of the binary phase-field-crystal model, Phys. Rev. E, 81, 1, 011602 (2010)
[9] Stefanovic, P.; Haataja, M.; Provatas, N., Phase field crystal study of deformation and plasticity in nanocrystalline materials, Phys. Rev. E, 80, 046107 (2009)
[10] Guan, Z.; Heinonen, V.; Lowengrub, J.; Wang, C.; Wise, S., An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations, J. Comput. Phys., 321, 1026-1054 (2016) · Zbl 1349.74360
[11] Swift, J.; Hohenberg, P., Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15, 319 (1977)
[12] Wang, C.; Wise, S., Global smooth solutions of the modified phase field crystal equation, Methods Appl. Anal., 17, 191-212 (2010) · Zbl 1217.35040
[13] Cheng, M.; Warren, J., An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227, 6241 (2008) · Zbl 1151.82411
[14] Vollmayr-Lee, B.; Rutenberg, A., Fast and accurate coarsening simulation with an unconditionally stable time step, Phys. Rev. E, 68, 066703 (2003)
[15] Tegze, G.; Bansel, G.; Tóth, G.; Pusztai, T.; Fan, Z.; Gránásy, L., Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients, J. Comput. Phys., 228, 1612-1623 (2009) · Zbl 1156.82373
[16] Athreya, B.; Goldenfeld, N.; Dantzig, J.; Greenwood, M.; Provatas, N., Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model, Phys. Rev. E, 76, 056706 (2007)
[17] Cao, H.; Sun, Z., Two finite difference schemes for the phase field crystal equation, Sci. China Math., 58, 11, 2435-2454 (2015) · Zbl 1335.82030
[18] Guo, R.; Xu, Y., Local discontinuous galerkin method and high order semi-implicit scheme for the phase field crystal equation, SIAM J. Sci. Comput., 38, 1, A105-A127 (2016) · Zbl 1330.65149
[19] Hirouchi, T.; Takaki, T.; Tomita, Y., Development of numerical scheme for phase field crystal deformation simulation, Comput. Mater. Sci., 44, 1192-1197 (2009)
[20] Eyre, D., Unconditionally gradient stable time marching the Cahn-Hilliard equation, (Bullard, J. W.; Kalia, R.; Stoneham, M.; Chen, L., Computational and Mathematical Models of Microstructural Evolution, Vol. 53 (1998), Materials Research Society: Materials Research Society Warrendale, PA, USA), 1686-1712
[21] Hu, Z.; Wise, S.; Wang, C.; Lowengrub, J., Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation, J. Comput. Phys., 228, 5323-5339 (2009) · Zbl 1171.82015
[22] Baskaran, A.; Hu, Z.; Lowengrub, J.; Wang, C.; Wise, S.; Zhou, P., Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, J. Comput. Phys., 250, 270-292 (2013) · Zbl 1349.65265
[23] Baskaran, A.; Lowengrub, J.; Wang, C.; Wise, S., Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 51, 2851-2873 (2013) · Zbl 1401.82046
[24] 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)
[25] Grasselli, M.; Pierre, M., Energy stable and convergent finite element schemes for the modified phase field crystal equation, ESAIM: M2AN, 50, 5, 1523-1560 (2016) · Zbl 1358.82025
[26] Wang, C.; Wise, S., An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49, 945-969 (2011) · Zbl 1230.82005
[27] Wise, S.; Wang, C.; Lowengrub, J., An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47, 2269-2288 (2009) · Zbl 1201.35027
[28] Christlieb, A.; Jones, J.; Promislow, K.; Wetton, B.; Willoughby, M., High accuracy solutions to energy gradient flows from material science models, J. Comput. Phys., 257, 193-215 (2014) · Zbl 1349.65510
[29] Guillén-González, F.; Tierra, G., Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68, 8, 821-846 (2014) · Zbl 1362.65104
[30] Yang, X.; Han, D., Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal equation, J. Comput. Phys., 330, 1116-1134 (2017) · Zbl 1380.65209
[31] Han, D.; Brylev, A.; Yang, X.; Tan, Z., Numerical analysis of second order, fully discrete energy stable schemes for phase field models of two phase incompressible flows, J. Sci. Comput., 1-25 (2016)
[32] Yang, X., Linear, and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 302, 509-523 (2016)
[33] Zhao, J.; Wang, Q.; Yang, X., Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach, Internat. J. Numer. Methods Engrg., 110, 3, 279-300 (2017) · Zbl 1365.74138
[34] Diegel, A.; Wang, C.; Wise, S., Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation, IMA J. Numer. Anal., 36, 1867-1897 (2016) · Zbl 1433.80005
[35] Guo, J.; Wang, C.; Wise, S.; Yue, X., An \(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation, Commun. Math. Sci., 14, 489-515 (2016) · Zbl 1338.65221
[36] 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
[37] Gomez, H.; Nogueira, X., An unconditionally energy-stable method for the phase field crystal equation, Comput. Methods Appl. Mech. Engrg., 249, 52-61 (2012) · Zbl 1348.74280
[38] 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
[39] Collins, C.; Shen, J.; Wise, S., An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system, Commun. Comput. Phys., 13, 929-957 (2013) · Zbl 1373.76161
[40] Guan, Z.; Lowengrub, J.; Wang, C.; Wise, S., Second-order convex splitting schemes for nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277, 48-71 (2014) · Zbl 1349.65298
[41] Guan, Z.; Wang, C.; Wise, S., A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128, 377-406 (2014) · Zbl 1304.65209
[42] Wise, S., Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, J. Sci. Comput., 44, 38-68 (2010) · Zbl 1203.76153
[43] Trefethen, L., Spectral Methods in MATLAB, Vol. 10 (2000), SIAM · Zbl 0953.68643
[44] Heywood, J. G.; Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for the second-order time discretization, SIAM J. Numer. Anal., 27, 353-384 (1990) · Zbl 0694.76014
[45] Trottenberg, U.; Oosterlee, C. W.; Schuller, A., Multigrid (2000), Academic Press
[46] W. Feng, Z. Guo, J. Lowengrub, S. Wise, Mass-conservative cell-centered finite difference methods and an efficient multigrid solver for the diffusion equation on block-structured, locally cartesian adaptive grids, 2016 (in preparation).; W. Feng, Z. Guo, J. Lowengrub, S. Wise, Mass-conservative cell-centered finite difference methods and an efficient multigrid solver for the diffusion equation on block-structured, locally cartesian adaptive grids, 2016 (in preparation). · Zbl 1380.65402
[47] W. Feng, Z. Guan, J. Lowengrub, S. Wise, C. Wang, An energy stable finite-difference scheme for functionalized Cahn-HilliardEquation and its convergence analysis, 2016, arXiv preprint arXiv:1610.02473; W. Feng, Z. Guan, J. Lowengrub, S. Wise, C. Wang, An energy stable finite-difference scheme for functionalized Cahn-HilliardEquation and its convergence analysis, 2016, arXiv preprint arXiv:1610.02473
[48] Feng, W.; Salgado, A.; Wang, C.; Wise, S., Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334, 45-67 (2017) · Zbl 1375.35149
[49] Kay, D.; Welford, R., A multigrid finite element solver for the Cahn-Hilliard equation, J. Comput. Phys., 212, 1, 288-304 (2006) · Zbl 1081.65091
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.