The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded from below restriction of the nonlinear free energy potential. Then, we prove the unconditional energy stability for the semi-discrete schemes carefully and rigorously. Another contribution is that we can discrete the auxiliary variable combined with the nonlinear term totally explicitly. Such modification is very efficient for fast calculation. Furthermore, the positive property of $r$ can be guaranteed which is very important and reasonable for the models' equivalence. Besides, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed E-SAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

[1]  Axel Voigt,et al.  Margination of white blood cells: a computational approach by a hydrodynamic phase field model , 2015, Journal of Fluid Mechanics.

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

[3]  P. Lin,et al.  A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects , 2014, Journal of Fluid Mechanics.

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

[5]  Wheeler,et al.  Phase-field model for isothermal phase transitions in binary alloys. , 1992, Physical review. A, Atomic, molecular, and optical physics.

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

[7]  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.

[8]  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.

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

[10]  J. Mixter.  Fast , 2012 .

[11]  Bruce T. Murray,et al.  Computation of Dendrites Using a Phase Field Model , 2017 .

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

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

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

[15]  Qiang Du,et al.  Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen-Cahn Equation , 2019, SIAM J. Numer. Anal..

[16]  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..

[17]  Xiao Li,et al.  Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation , 2018, J. Comput. Phys..

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

[19]  Xiaoli Li,et al.  Efficient modified techniques of invariant energy quadratization approach for gradient flows , 2019, Appl. Math. Lett..

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

[21]  Jaemin Shin,et al.  First and second order numerical methods based on a new convex splitting for phase-field crystal equation , 2016, J. Comput. Phys..

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

[23]  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..

[24]  Christian Miehe,et al.  A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits , 2010 .

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

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

[27]  Waixiang Cao,et al.  An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection , 2018, Comput. Phys. Commun..

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

[29]  Qiang Du,et al.  Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models , 2016, J. Comput. Phys..

[30]  Laura De Lorenzis,et al.  A review on phase-field models of brittle fracture and a new fast hybrid formulation , 2015 .

[31]  Xiaofeng Yang,et al.  Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System , 2017, Journal of Scientific Computing.

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

[33]  Yibao Li,et al.  An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation , 2017 .

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