Energy stability and convergence of SAV block-centered finite difference method for gradient flows

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

[1]  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.

[2]  Xiaobing Feng,et al.  Fully Discrete Finite Element Approximations of the Navier-Stokes-Cahn-Hilliard Diffuse Interface Model for Two-Phase Fluid Flows , 2006, SIAM J. Numer. Anal..

[3]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[4]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[5]  Jun Li,et al.  Energy Stable Numerical Schemes for a Hydrodynamic Model of Nematic Liquid Crystals , 2016, SIAM J. Sci. Comput..

[6]  Steven M. Wise,et al.  Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System , 2013, SIAM J. Numer. Anal..

[7]  Steven M. Wise,et al.  Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method , 2007, J. Comput. Phys..

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

[9]  G. Grün,et al.  On Convergent Schemes for Diffuse Interface Models for Two-Phase Flow of Incompressible Fluids with General Mass Densities , 2013, SIAM J. Numer. Anal..

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

[11]  Jie Shen,et al.  Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows , 2018, SIAM J. Numer. Anal..

[12]  Xiaofeng Yang,et al.  Dynamics of Defect Motion in Nematic Liquid Crystal Flow: Modeling and Numerical Simulation , 2007 .

[13]  Xiaofeng Yang,et al.  A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals , 2017 .

[14]  Cheng Wang,et al.  Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation , 2015, Math. Comput..

[15]  Charles M. Elliott,et al.  CONVERGENCE OF NUMERICAL SOLUTIONS TO THE ALLEN-CAHN EQUATION , 1998 .

[16]  Ying Chen,et al.  Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models , 2016, J. Comput. Phys..

[17]  Xiaofeng Yang,et al.  Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends , 2016, J. Comput. Phys..

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

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

[20]  Mary F. Wheeler,et al.  A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations , 1998 .

[21]  Jie Shen,et al.  A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows with Different Densities and Viscosities , 2010, SIAM J. Sci. Comput..

[22]  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..

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

[24]  Charles M. Elliott,et al.  A second order splitting method for the Cahn-Hilliard equation , 1989 .

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

[26]  James J. Feng,et al.  A diffuse-interface method for simulating two-phase flows of complex fluids , 2004, Journal of Fluid Mechanics.

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

[28]  Jie Shen,et al.  A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows , 2017, SIAM Rev..

[29]  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 .

[30]  Xiaofeng Yang,et al.  Numerical approximations of the Cahn-Hilliard and Allen-Cahn Equations with general nonlinear potential using the Invariant Energy Quadratization approach , 2017, 1712.02760.

[31]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

[32]  Andreas Prohl,et al.  Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows , 2003, Numerische Mathematik.

[33]  Andreas Prohl,et al.  Error analysis of a mixed finite element method for the Cahn-Hilliard equation , 2004, Numerische Mathematik.

[34]  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..