Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation

Abstract 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.

[1]  Axel Voigt,et al.  A phase field crystal study of heterogeneous nucleation – application of the string method , 2014 .

[2]  Zhi-zhong Sun,et al.  Two finite difference schemes for the phase field crystal equation , 2015 .

[3]  Steven M. Wise,et al.  Unconditionally stable schemes for equations of thin film epitaxy , 2010 .

[4]  Cheng Wang,et al.  Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation , 2012, J. Comput. Phys..

[5]  Nikolas Provatas,et al.  Phase field crystal study of deformation and plasticity in nanocrystalline materials. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Axel Voigt,et al.  Nucleation and growth by a phase field crystal (PFC) model , 2007 .

[7]  Steven M. Wise,et al.  An Energy Stable and Convergent Finite-Difference Scheme for the Modified Phase Field Crystal Equation , 2011, SIAM J. Numer. Anal..

[8]  E Granato,et al.  Dynamical transitions and sliding friction of the phase-field-crystal model with pinning. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  D. J. Eyre.  Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation , 1998 .

[10]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

[11]  A. Voigt,et al.  A Navier-Stokes phase-field crystal model for colloidal suspensions. , 2013, The Journal of chemical physics.

[12]  Axel Voigt,et al.  A Phase Field Crystal Approach for Particles in a Flowing Solvent , 2011 .

[13]  Cheng Wang,et al.  An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation , 2009, SIAM J. Numer. Anal..

[14]  Badrinarayan P. Athreya,et al.  Renormalization Group Approach to Multiscale Modelling in Materials Science , 2005, cond-mat/0508671.

[15]  Badrinarayan P. Athreya,et al.  Renormalization-group theory for the phase-field crystal equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Yan Xu,et al.  Local Discontinuous Galerkin Method and High Order Semi-Implicit Scheme for the Phase Field Crystal Equation , 2016, SIAM J. Sci. Comput..

[17]  T Ala-Nissila,et al.  Phase-field-crystal models and mechanical equilibrium. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Tomohiro Takaki,et al.  Development of numerical scheme for phase field crystal deformation simulation , 2009 .

[19]  Yuan Ma,et al.  An adaptive time-stepping strategy for solving the phase field crystal model , 2013, J. Comput. Phys..

[20]  Amanda E. Diegel,et al.  Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation , 2014, 1411.5248.

[21]  James A. Warren,et al.  An efficient algorithm for solving the phase field crystal model , 2008, J. Comput. Phys..

[22]  Badrinarayan P. Athreya,et al.  Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  László Gránásy,et al.  Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients , 2009, J. Comput. Phys..

[24]  Cheng Wang,et al.  Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms , 2016, J. Comput. Phys..

[25]  Alain Karma,et al.  Phase-field crystal study of grain-boundary premelting , 2008, 0807.5083.

[26]  Xiaofeng Yang,et al.  Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach , 2017 .

[27]  Zhi-Feng Huang,et al.  Density-amplitude formulation of the phase-field crystal model for two-phase coexistence in two and three dimensions , 2010 .

[28]  Keith Promislow,et al.  High accuracy solutions to energy gradient flows from material science models , 2014, J. Comput. Phys..

[29]  Cheng Wang,et al.  Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation , 2009, J. Comput. Phys..

[30]  Axel Voigt,et al.  Particles at fluid-fluid interfaces: A new Navier-Stokes-Cahn-Hilliard surface- phase-field-crystal model. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  B. Vollmayr-Lee,et al.  Fast and accurate coarsening simulation with an unconditionally stable time step. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  E Granato,et al.  Thermal fluctuations and phase diagrams of the phase-field crystal model with pinning. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  N. Provatas,et al.  Phase-field crystals with elastic interactions. , 2006, Physical review letters.

[34]  P. Voorhees,et al.  Controlling crystal symmetries in phase-field crystal models , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[35]  Daozhi Han,et al.  Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model , 2017, J. Comput. Phys..

[36]  M. Dehghan,et al.  The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods , 2016 .

[37]  Cheng Wang,et al.  Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations , 2014, J. Comput. Phys..

[38]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[39]  Joel Berry,et al.  Melting at dislocations and grain boundaries: A phase field crystal study , 2008, 0801.0765.

[40]  Steven M. Wise,et al.  Convergence Analysis of a Second Order Convex Splitting Scheme for the Modified Phase Field Crystal Equation , 2012, SIAM J. Numer. Anal..

[41]  Peter W Voorhees,et al.  Stress-induced morphological instabilities at the nanoscale examined using the phase field crystal approach , 2009 .

[42]  A. A. Wheeler,et al.  Phase-field theory of edges in an anisotropic crystal , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[43]  Pedro Tarazona,et al.  Dynamic density functional theory of fluids , 1999 .

[44]  Cheng Wang,et al.  An Energy Stable Finite-Difference Scheme for Functionalized Cahn-Hilliard Equation and its Convergence Analysis , 2016, 1610.02473.

[45]  Joel Berry,et al.  Modeling multiple time scales during glass formation with phase-field crystals. , 2011, Physical review letters.

[46]  M. Grasselli,et al.  Energy stable and convergent finite element schemes for the modified phase field crystal equation , 2016 .

[47]  Cheng Wang,et al.  An $H^2$ convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn–Hilliard equation , 2016 .

[48]  Daozhi Han,et al.  Numerical Analysis of Second Order, Fully Discrete Energy Stable Schemes for Phase Field Models of Two-Phase Incompressible Flows , 2017, J. Sci. Comput..

[49]  Rolf Rannacher,et al.  On the finite element approximation of the nonstationary Navier-Stokes problem , 1980 .

[50]  Xesús Nogueira,et al.  An unconditionally energy-stable method for the phase field crystal equation , 2012 .

[51]  Badrinarayan P. Athreya,et al.  Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  M. Grant,et al.  Phase-field crystal modeling and classical density functional theory of freezing , 2007 .

[53]  S. M. Wise,et al.  Unconditionally Stable Finite Difference, Nonlinear Multigrid Simulation of the Cahn-Hilliard-Hele-Shaw System of Equations , 2010, J. Sci. Comput..

[54]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[55]  M. Grant,et al.  Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  E Granato,et al.  Nonlinear driven response of a phase-field crystal in a periodic pinning potential. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Francisco Guillén-González,et al.  Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models , 2014, Comput. Math. Appl..

[58]  Wenqiang Feng,et al.  A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids , 2018, J. Comput. Phys..

[59]  Richard Welford,et al.  A multigrid finite element solver for the Cahn-Hilliard equation , 2006, J. Comput. Phys..

[60]  Xiaofeng Yang,et al.  Decoupled energy stable schemes for phase-field vesicle membrane model , 2015, J. Comput. Phys..

[61]  Ilya Starodumov,et al.  Three dimensional structures predicted by the modified phase field crystal equation , 2016 .

[62]  Robert Spatschek,et al.  Amplitude equations for polycrystalline materials with interaction between composition and stress , 2010, 1002.1580.

[63]  Cheng Wang,et al.  An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations , 2016, J. Comput. Phys..

[64]  Cheng Wang,et al.  A Linear Energy Stable Scheme for a Thin Film Model Without Slope Selection , 2012, J. Sci. Comput..

[65]  Badrinarayan P. Athreya,et al.  Using the phase-field crystal method in the multi-scale modeling of microstructure evolution , 2007 .

[66]  Cheng Wang,et al.  A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation , 2014, Numerische Mathematik.

[67]  Jie Shen,et al.  An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System , 2013 .

[68]  Cheng Wang,et al.  A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations , 2016, Discrete & Continuous Dynamical Systems - B.

[69]  Jie Shen,et al.  Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy , 2012, SIAM J. Numer. Anal..

[70]  E Granato,et al.  Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[71]  N. Provatas,et al.  Amplitude expansion of the binary phase-field-crystal model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  Martin Grant,et al.  Modeling elasticity in crystal growth. , 2001, Physical review letters.