A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

In this article, we study a new second‐order energy stable Backward Differentiation Formula (BDF) finite difference scheme for the epitaxial thin film equation with slope selection (SS). One major challenge for higher‐order‐in‐time temporal discretizations is how to ensure an unconditional energy stability without compromising numerical efficiency or accuracy. We propose a framework for designing a second‐order numerical scheme with unconditional energy stability using the BDF method with constant coefficient stabilizing terms. Based on the unconditional energy stability property that we establish, we derive an ℓ ∞ ( 0 , T ; H h 2 ) stability for the numerical solution and provide an optimal convergence analysis. To deal with the highly nonlinear four‐Laplacian term at each time step, we apply efficient preconditioned steepest descent and preconditioned nonlinear conjugate gradient algorithms to solve the corresponding nonlinear system. Various numerical simulations are presented to demonstrate the stability and efficiency of the proposed schemes and solvers. Comparisons with other second‐order schemes are presented.

[1]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

[2]  E. Polak,et al.  Note sur la convergence de méthodes de directions conjuguées , 1969 .

[3]  Yoshikazu Giga,et al.  A mathematical problem related to the physical theory of liquid crystal configurations , 1987 .

[4]  Schick,et al.  Correlation between structural and interfacial properties of amphiphilic systems. , 1990, Physical review letters.

[5]  R. Nicolaides,et al.  Numerical analysis of a continuum model of phase transition , 1991 .

[6]  M. Ortiz,et al.  The morphology and folding patterns of buckling-driven thin-film blisters , 1994 .

[7]  M. Ortiz,et al.  Delamination of Compressed Thin Films , 1997 .

[8]  Camillo De Lellis,et al.  Line energies for gradient vector fields in the plane , 1999 .

[9]  Bo Li,et al.  Center for Scientific Computation And Mathematical Modeling , 2003 .

[10]  A. Nepomnyashchy,et al.  Disclinations in square and hexagonal patterns. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Robert V. Kohn,et al.  Upper bound on the coarsening rate for an epitaxial growth model , 2003 .

[12]  Bo Li,et al.  Epitaxial Growth Without Slope Selection: Energetics, Coarsening, and Dynamic Scaling , 2004, J. Nonlinear Sci..

[13]  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.

[14]  Yunqing Huang,et al.  PRECONDITIONED HYBRID CONJUGATE GRADIENT ALGORITHM FOR P-LAPLACIAN , 2005 .

[15]  Tao Tang,et al.  Stability Analysis of Large Time-Stepping Methods for Epitaxial Growth Models , 2006, SIAM J. Numer. Anal..

[16]  Robert V. Kohn,et al.  Energy-Driven Pattern Formation , 2006 .

[17]  Alan R. Champneys,et al.  Localized Hexagon Patterns of the Planar Swift-Hohenberg Equation , 2008, SIAM J. Appl. Dyn. Syst..

[18]  Radosław Pytlak,et al.  Conjugate Gradient Algorithms in Nonconvex Optimization , 2008 .

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

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

[21]  Tao Tang,et al.  An Adaptive Time-Stepping Strategy for the Molecular Beam Epitaxy Models , 2011, SIAM J. Sci. Comput..

[22]  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..

[23]  Keith Promislow,et al.  Curvature driven flow of bi-layer interfaces , 2011 .

[24]  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..

[25]  Keith Promislow,et al.  Variational Models of Network Formation and Ion Transport: Applications to Perfluorosulfonate Ionomer Membranes , 2012 .

[26]  Zhengru Zhang,et al.  The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model , 2012 .

[27]  Shibin Dai,et al.  Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Keith Promislow,et al.  Meander and Pearling of Single-Curvature Bilayer Interfaces in the Functionalized Cahn-Hilliard Equation , 2014, SIAM J. Math. Anal..

[29]  Cheng Wang,et al.  A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection , 2014, J. Sci. Comput..

[30]  Jose A. Moreno-Cadenas,et al.  Formation of square patterns using a model alike Swift-Hohenberg , 2014, 2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE).

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

[32]  Keith Promislow,et al.  Existence of pearled patterns in the planar functionalized Cahn–Hilliard equation , 2014, 1410.0447.

[33]  K. Promislow,et al.  Geometric Evolution of Quasi-Bilayers in Multicomponent Functionalized Cahn-Hilliard Equation , 2015 .

[34]  Tao Tang,et al.  Fast and stable explicit operator splitting methods for phase-field models , 2015, J. Comput. Phys..

[35]  Zhonghua Qiao,et al.  Convergence of a Fast Explicit Operator Splitting Method for the Molecular Beam Epitaxy Model , 2015, 1506.05271.

[36]  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 .

[37]  Zhonghua Qiao,et al.  Characterizing the Stabilization Size for Semi-Implicit Fourier-Spectral Method to Phase Field Equations , 2014, SIAM J. Numer. Anal..

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

[39]  K. Promislow,et al.  Existence, bifurcation, and geometric evolution of quasi-bilayers in the multicomponent functionalized Cahn–Hilliard equation , 2015, Journal of Mathematical Biology.

[40]  Jaemin Shin,et al.  A Second-Order Operator Splitting Fourier Spectral Method for Models of Epitaxial Thin Film Growth , 2017, J. Sci. Comput..

[41]  Dong Li,et al.  On Second Order Semi-implicit Fourier Spectral Methods for 2D Cahn–Hilliard Equations , 2017, J. Sci. Comput..

[42]  Hui Zhang,et al.  Convergence of a Fast Explicit Operator Splitting Method for the Epitaxial Growth Model with Slope Selection , 2017, SIAM J. Numer. Anal..

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

[44]  Jia Zhao,et al.  Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method , 2017, J. Comput. Phys..

[45]  Cheng Wang,et al.  A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection , 2018, J. Sci. Comput..

[46]  Cheng Wang,et al.  A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation , 2018 .

[47]  Wenqiang Feng,et al.  Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation , 2016, Comput. Math. Appl..

[48]  Jiang Yang,et al.  The scalar auxiliary variable (SAV) approach for gradient flows , 2018, J. Comput. Phys..

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