The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems

The invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches are two recently proposed methods to develop linear and unconditionally energy stable schemes for a class of dissipative/conservative systems. The essential idea of these two methods is the energy quadratization strategy, where either the nonlinear potential or its integral is transformed into quadratic forms of the new auxiliary variables. We present the IEQ and SAV approaches in a unified and more general setting, show a few typical applications to problems with moderately stiff nonlinearities, and then present the stabilized IEQ and SAV approaches to deal with several complex systems with highly stiff nonlinear terms.

[1]  Jie Shen,et al.  Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose-Einstein Condensates , 2019, J. Comput. Phys..

[2]  Jie Shen,et al.  Efficient energy stable schemes with spectral discretization in space for anisotropic , 2013 .

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

[4]  Hui Zhang,et al.  Efficient and linear schemes for anisotropic Cahn-Hilliard model using the Stabilized-Invariant Energy Quadratization (S-IEQ) approach , 2019, Comput. Phys. Commun..

[5]  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..

[6]  Jun Zhang,et al.  On efficient numerical schemes for a two-mode phase field crystal model with face-centered-cubic (FCC) ordering structure , 2019 .

[7]  Jia Zhao,et al.  A GENERAL STRATEGY FOR NUMERICAL APPROXIMATIONS OF NON-EQUILIBRIUM MODELS – PART I : THERMODYNAMICAL SYSTEMS , 2018 .

[8]  Q. Du,et al.  The phase field method for geometric moving interfaces and their numerical approximations , 2019, Geometric Partial Differential Equations - Part I.

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

[10]  A. A. Wheeler,et al.  Thermodynamically-consistent phase-field models for solidification , 1992 .

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

[12]  Xiaoming He,et al.  Numerical approximations for the hydrodynamics coupled binary surfactant phase field model: Second-order, linear, unconditionally energy-stable schemes , 2019, Communications in Mathematical Sciences.

[13]  Jie Shen,et al.  Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs , 2020, J. Comput. Phys..

[14]  H. Gómez,et al.  Computational Phase‐Field Modeling , 2017 .

[15]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[16]  Franck Boyer,et al.  Study of a three component Cahn-Hilliard flow model , 2006 .

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

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

[19]  van der Kg Kristoffer Zee,et al.  Stabilized second‐order convex splitting schemes for Cahn–Hilliard models with application to diffuse‐interface tumor‐growth models , 2014, International journal for numerical methods in biomedical engineering.

[20]  Feng Chen,et al.  Efcient Energy Stable Schemes with Spectral Discretization in Space for Anisotropic Cahn-Hilliard Systems , 2020 .

[21]  Franck Boyer,et al.  Numerical schemes for a three component Cahn-Hilliard model , 2011 .

[22]  Chaolong Jiang,et al.  A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach , 2018, Journal of Scientific Computing.

[23]  Lili Ju,et al.  Linear and unconditionally energy stable schemes for the binary fluid–surfactant phase field model , 2017, 1701.07446.

[24]  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.

[25]  Shuyu Sun,et al.  Linearly Decoupled Energy-Stable Numerical Methods for Multicomponent Two-Phase Compressible Flow , 2018, SIAM J. Numer. Anal..

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

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

[28]  Axel Voigt,et al.  A new phase-field model for strongly anisotropic systems , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[30]  Jun Zhang,et al.  Efficient second order unconditionally stable time marching numerical scheme for a modified phase-field crystal model with a strong nonlinear vacancy potential , 2019, Comput. Phys. Commun..

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

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

[33]  Qing Cheng,et al.  Multiple Scalar Auxiliary Variable (MSAV) Approach and its Application to the Phase-Field Vesicle Membrane Model , 2018, SIAM J. Sci. Comput..

[34]  S. Dong,et al.  An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices , 2018, J. Comput. Phys..

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

[36]  A. Karma,et al.  Phase-field model of dendritic sidebranching with thermal noise. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  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.

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

[39]  Xiaofeng Yang,et al.  Efficient numerical scheme for a dendritic solidification phase field model with melt convection , 2019, J. Comput. Phys..

[40]  Collins,et al.  Diffuse interface model of diffusion-limited crystal growth. , 1985, Physical review. B, Condensed matter.

[41]  Jian Zhang,et al.  Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations , 2015, J. Sci. Comput..

[42]  Xiaofeng Yang,et al.  Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method , 2017 .

[43]  Alain Karma,et al.  Phase-field-crystal model for fcc ordering. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  P. C. Hohenberg,et al.  Renormalization-group methods for critical dynamics: I. Recursion relations and effects of energy conservation , 1974 .

[45]  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..

[46]  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..

[47]  Xiaofeng Yang,et al.  Efficient linear, stabilized, second-order time marching schemes for an anisotropic phase field dendritic crystal growth model , 2018, Computer Methods in Applied Mechanics and Engineering.

[48]  Suchuan Dong,et al.  Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable , 2018, J. Comput. Phys..

[49]  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..

[50]  Lili Ju,et al.  Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model , 2017 .

[51]  R. Kobayashi.  Modeling and numerical simulations of dendritic crystal growth , 1993 .

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

[53]  Jie Shen,et al.  Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows , 2020, Numerische Mathematik.

[54]  Jia Zhao,et al.  Efficient linear schemes for the nonlocal Cahn-Hilliard equation of phase field models , 2019, Comput. Phys. Commun..

[55]  N. Goldenfeld,et al.  Molecular dynamics on diffusive time scales from the phase-field-crystal equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Xiaofeng Yang,et al.  Convergence Analysis for the Invariant Energy Quadratization (IEQ) Schemes for Solving the Cahn–Hilliard and Allen–Cahn Equations with General Nonlinear Potential , 2020, J. Sci. Comput..

[57]  Xiaofeng Yang,et al.  Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard Model , 2019, Computer Methods in Applied Mechanics and Engineering.

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

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

[60]  Jie Shen,et al.  Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows , 2015, SIAM J. Numer. Anal..

[61]  Jie Shen,et al.  Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model , 2017, J. Comput. Phys..

[62]  Ignacio Romero,et al.  Thermodynamically consistent time‐stepping algorithms for non‐linear thermomechanical systems , 2009 .

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

[64]  Xiaofeng Yang,et al.  A novel decoupled and stable scheme for an anisotropic phase-field dendritic crystal growth model , 2019, Appl. Math. Lett..

[65]  Jie Shen,et al.  Energy stability and convergence of SAV block-centered finite difference method for gradient flows , 2018, Math. Comput..

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

[67]  Jie Shen,et al.  Highly Efficient and Accurate Numerical Schemes for the Epitaxial Thin Film Growth Models by Using the SAV Approach , 2019, J. Sci. Comput..