Efficient and accurate structure preserving schemes for complex nonlinear systems

Abstract The scalar auxiliary variable (SAV) approach was recently proposed in Shen et al. (2018) to construct efficient and accurate unconditionally energy stable schemes for gradient flows. We present the SAV approach in a more general setting, and consider a few interesting extensions that enable us to use the SAV approach to deal with a large class of complex dissipative/conservative systems.

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

[2]  M. Gurtin,et al.  TWO-PHASE BINARY FLUIDS AND IMMISCIBLE FLUIDS DESCRIBED BY AN ORDER PARAMETER , 1995, patt-sol/9506001.

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

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

[5]  S. Edwards,et al.  The Theory of Polymer Dynamics , 1986 .

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

[7]  Shuyu Sun,et al.  Decoupled, energy stable schemes for a phase-field surfactant model , 2018, Comput. Phys. Commun..

[8]  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.

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

[10]  R. Larson.  Arrested Tumbling in Shearing Flows of Liquid Crystal Polymers , 1990 .

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

[12]  M. Gregory Forest,et al.  The weak shear kinetic phase diagram for nematic polymers , 2004 .

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

[14]  Felix Otto,et al.  Lubrication approximation with prescribed nonzero contact anggle , 1998 .

[15]  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..

[16]  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..

[17]  Ricardo H. Nochetto,et al.  A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality , 2004 .

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

[19]  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.

[20]  Xiaofeng Yang,et al.  Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows , 2010 .

[21]  Zhangxin Chen,et al.  Efficient Linearly and Unconditionally Energy Stable Schemes for the Phase Field Model of Solid-State Dewetting Problems , 2018, ICCS.

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

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

[24]  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..

[25]  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..

[26]  Lorenzo Giacomelli,et al.  Variatonal formulation for the lubrication approximation of the Hele-Shaw flow , 2001 .

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

[28]  Yongzhong Song,et al.  Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions , 2018, J. Comput. Phys..

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

[30]  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 .

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

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

[33]  M. Gregory Forest,et al.  The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: finite shear rates , 2004 .

[34]  F. M. Leslie.  Theory of Flow Phenomena in Liquid Crystals , 1979 .

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

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

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

[38]  Desai,et al.  Numerical study of late-stage coarsening for off-critical quenches in the Cahn-Hilliard equation of phase separation. , 1989, Physical review. B, Condensed matter.

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

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

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

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

[43]  Xiaofeng Yang,et al.  Efficient schemes with unconditionally energy stability for the anisotropic Cahn-Hilliard Equation using the stabilized-Scalar Augmented Variable (S-SAV) approach , 2018, 1804.02619.

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

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

[46]  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..

[47]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .