On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $\alpha$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-\alpha/3$ power law coarsening stage.

[1]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[2]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[3]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[4]  G. Karniadakis,et al.  A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations , 2016 .

[5]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .

[6]  Tao Tang,et al.  A finite difference scheme for partial integro-differential equations with a weakly singular kernel , 1993 .

[7]  Endre Süli,et al.  Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth , 2010, Math. Comput..

[8]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[9]  Mark Ainsworth,et al.  Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation , 2017 .

[10]  Wouter-Jan Rappel,et al.  Computational model for cell morphodynamics. , 2010, Physical review letters.

[11]  Hong Wang,et al.  On power law scaling dynamics for time-fractional phase field models during coarsening , 2018, Commun. Nonlinear Sci. Numer. Simul..

[12]  Xianmin Xu,et al.  Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines , 2017, Journal of Fluid Mechanics.

[13]  Suchuan Dong,et al.  On imposing dynamic contact-angle boundary conditions for wall-bounded liquid–gas flows , 2012 .

[14]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[15]  Haijun Yu,et al.  Energy-stable second-order linear schemes for the Allen–Cahn phase-field equation , 2018, Communications in Mathematical Sciences.

[16]  Xiaofeng Yang,et al.  Efficient Second Order Unconditionally Stable Schemes for a Phase Field Moving Contact Line Model Using an Invariant Energy Quadratization Approach , 2018, SIAM J. Sci. Comput..

[17]  Giulio Schimperna,et al.  Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations , 2015, 1502.06383.

[18]  Kristoffer G. van der Zee,et al.  Numerical simulation of a thermodynamically consistent four‐species tumor growth model , 2012, International journal for numerical methods in biomedical engineering.

[19]  Cheng Wang,et al.  Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system , 2016, Numerische Mathematik.

[20]  Hong Wang,et al.  A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation , 2017, J. Comput. Phys..

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

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

[23]  Bangti Jin,et al.  Numerical Analysis of Nonlinear Subdiffusion Equations , 2017, SIAM J. Numer. Anal..

[24]  Qing Nie,et al.  Nonlinear three-dimensional simulation of solid tumor growth , 2007 .

[25]  Andreas Prohl,et al.  Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part I: Error analysis under minimum regularities , 2001 .

[26]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[27]  Clarke,et al.  Origin of reflection high-energy electron-diffraction intensity oscillations during molecular-beam epitaxy: A computational modeling approach. , 1987, Physical review letters.

[28]  Alexander A. Nepomnyashchy,et al.  Front-type solutions of fractional Allen–Cahn equation , 2008 .

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

[30]  Christian Kahle,et al.  Preconditioning of a coupled Cahn--Hilliard Navier--Stokes system , 2016, 1610.03991.

[31]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[32]  Q. Du,et al.  A phase field approach in the numerical study of the elastic bending energy for vesicle membranes , 2004 .

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

[34]  Charles M. Elliott,et al.  The global dynamics of discrete semilinear parabolic equations , 1993 .

[35]  Jie Shen,et al.  On the maximum principle preserving schemes for the generalized Allen–Cahn equation , 2016 .

[36]  Junseok Kim.  Phase-Field Models for Multi-Component Fluid Flows , 2012 .

[37]  J. Nohel,et al.  Frequency domain methods for Volterra equations , 1976 .

[38]  Qiang Du,et al.  Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods , 2019, Journal of Scientific Computing.

[39]  Zhiping Mao,et al.  Analysis and Approximation of a Fractional Cahn-Hilliard Equation , 2017, SIAM J. Numer. Anal..

[40]  Lin Wang,et al.  On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation , 2017, J. Sci. Comput..

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

[42]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

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

[44]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[45]  Jiwei Zhang,et al.  A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems , 2018, SIAM J. Numer. Anal..

[46]  Khader M. Hamdia,et al.  Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling , 2015 .

[47]  Thomas J. R. Hughes,et al.  Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models , 2011, J. Comput. Phys..

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

[49]  K ASSEM,et al.  Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations , 2014 .

[50]  Jie Shen,et al.  Efficient energy stable numerical schemes for a phase field moving contact line model , 2015, J. Comput. Phys..

[51]  L. Marino,et al.  The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids , 2013, Journal of Fluid Mechanics.

[52]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

[53]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[54]  Raytcho D. Lazarov,et al.  Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations , 2012, SIAM J. Numer. Anal..

[55]  Xiaofeng Yang,et al.  Numerical Approximations for Allen-Cahn Type Phase Field Model of Two-Phase Incompressible Fluids with Moving Contact Lines , 2017 .

[56]  Alexis Vasseur,et al.  A Parabolic Problem with a Fractional Time Derivative , 2015, 1501.07211.

[57]  Zhimin Zhang,et al.  Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations , 2015, 1511.03453.

[58]  Ping Sheng,et al.  Molecular scale contact line hydrodynamics of immiscible flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  Xiaofeng Yang,et al.  Numerical approximations for a phase-field moving contact line model with variable densities and viscosities , 2017, J. Comput. Phys..

[60]  D. Schötzau,et al.  Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations , 2014 .

[61]  William McLean,et al.  Numerical Solution of the Time-Fractional Fokker-Planck Equation with General Forcing , 2015, SIAM J. Numer. Anal..

[62]  H. Frieboes,et al.  Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method. , 2008, Journal of theoretical biology.

[63]  Pingwen Zhang,et al.  A Nonhomogeneous Kinetic Model of Liquid Crystal Polymers and Its Thermodynamic Closure Approximation , 2009 .

[64]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[65]  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 .

[66]  Haijun Yu,et al.  Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation , 2017, Journal of Mathematical Study.

[67]  Luis A. Caffarelli,et al.  An L∞ bound for solutions of the Cahn-Hilliard equation , 1995 .

[68]  Jia Zhao,et al.  A 3D Multi-Phase Hydrodynamic Model for Cytokinesis of Eukaryotic Cells , 2016 .