Frontiers of Mathematics in China >
Fast algorithm for viscous Cahn-Hilliard equation
Received date: 07 Mar 2020
Accepted date: 25 Aug 2021
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The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh (TT-M) finite element (FE) method to ease the problem caused by strong nonlinearities. The TT-M FE algorithm includes the following main computing steps. First, a nonlinear FE method is applied on a coarse time-mesh τc. Here, the FE method is used for spatial discretization and the implicit second-order θ scheme (containing both implicit Crank-Nicolson and second-order backward difference) is used for temporal discretization. Second, based on the chosen initial iterative value, a linearized FE system on time fine mesh is solved, where some useful coarse numerical solutions are found by Lagrange’s interpolation formula. The analysis for both stability and a priori error estimates is made in detail. Numerical examples are given to demonstrate the validity of the proposed algorithm. Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time.
Danxia WANG , Yaqian LI , Xingxing WANG , Hongen JIA . Fast algorithm for viscous Cahn-Hilliard equation[J]. Frontiers of Mathematics in China, 2022 , 17(4) : 689 -713 . DOI: 10.1007/s11464-021-0974-x
1 |
Ayuso B , García-Archilla B , Novo J . The postprocessed mixed finite element method for the Navier-Stokes equations. SIAM J Numer Anal, 2005, 43 (3): 1091- 1111
|
2 |
Bertozzi A L , Esedoglu S , Gillette A . Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans Image Process, 2006, 16 (1): 285- 291
|
3 |
Cahn J W . Free energy of a nonuniform system II: Thermodynamic basis. J Chem Phys, 1959, 30 (5): 1121- 1124
|
4 |
Cahn J W , Hilliard J E . Free energy of a nonuniform system I: Interfacial free energy. J Chem Phys, 1958, 28 (2): 258- 267
|
5 |
Cahn J W , Hilliard J E . Free energy of a nonuniform system III: Nucleation in a two component incompressible fluid. J Chem Phys, 1959, 31 (3): 688- 699
|
6 |
Carolan D , Chong H M , Ivankovic A , Kinloch A J , Taylor A C . Co-continuous polymer systems: A numerical investigation. Comp Mater Sci, 2015, 98: 24- 33
|
7 |
Chen C J , Li K , Chen Y P , Huang Y Q . Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations. Adv Comput Math, 2019, 45: 611- 630
|
8 |
Choksi R , Peletier M A , Williams J F . On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. SIAM J Appl Math, 2009, 69 (6): 1712- 1738
|
9 |
Elliott C M , Stuart A M . Viscous Cahn-Hilliard equation II. Analysis. J Differential Equations, 1996, 128 (2): 387- 414
|
10 |
Galenko P . Phase-field models with relaxation of the diffusion flux in nonequilibrium solidification of a binary system. Phys Lett A, 2001, 287 (3-4): 190- 197
|
11 |
Galenko P , Jou D . Diffuse-interface model for rapid phase transformations in nonequilibrium systems.. Phys Rev E, 2005, 71 (4 Pt 2): 046125
|
12 |
Galenko P , Jou D . Kinetic contribution to the fast spinodal decomposition controlled by diffusion. Phys A, 2009, 388 (15-16): 3113- 3123
|
13 |
Galenko P , Lebedev V . Analysis of the dispersion relation in spinodal decomposition of a binary system. Phil Mag Lett, 2007, 87 (11): 821- 827
|
14 |
Galenko P , Lebedev V . Local nonequilibrium effect on spinodal decomposition in a binary system. Int J Thermophys, 2008, 11 (1): 21- 28
|
15 |
Galenko P , Lebedev V . Non-equilibrium effects in spinodal decomposition of a binary system. Phys Lett A, 2008, 372 (7): 985- 989
|
16 |
Gao G H , Sun H W , Sun Z Z . Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J Comput Phys, 2015, 280: 510- 528
|
17 |
He Y N , Liu Y X , Tang T . On large time-stepping methods for the Cahn-Hilliard equation. Appl Numer Math, 2007, 57(5-7): 616- 628
|
18 |
Hecht F , Pironneau O , Ohtsuka K . FreeFEM++. 2010,
|
19 |
Heida M . On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system. Internat J Engrg Sci, 2013, 62 (1): 126- 156
|
20 |
Ju L L , Zhang J , Du Q . Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations. Comput Mater Sci, 2015: 272- 282
|
21 |
Kania M B . Upper semicontinuity of global attractors for the perturbed viscous CahnHilliard equations. Topol Methods Nonlinear Anal, 2008, 32 (2): 327- 345
|
22 |
Layton W , Tobiska L . A two-level method with backtracking for the Navier-Stokes equations. SIAM J Numer Anal, 1998, 35 (5): 2035- 2054
|
23 |
Lecoq N , Zapolsky H , Galenko P . Evolution of the structure factor in a hyperbolic model of spinodal decomposition. Eur Phys J Spec Top, 2009, 177 (1): 165- 175
|
24 |
Li Y B , Choi J I , Kim J . A phase-field fluid modeling and computation with interfacial profile correction term.. Commun Nonlinear Sci Numer Simul, 2016, 30 (1-3): 84- 100
|
25 |
Li Y B , Choi J I , Kim J . Multi-component Cahn-Hilliard system with different boundary conditions in complex domains. J Comput Phys, 2016, 323: 1- 16
|
26 |
Li Y B , Shin J , Choi Y , Kim J . Three-dimensional volume reconstruction from slice data using phase-field models. Comput Vis Image Underst, 2015, 137: 115- 124
|
27 |
Liu Q F , Hou Y R , Wang Z H , Zhao J K . Two-level methods for the Cahn-Hilliard equation. Math Comput Simulation, 2016, 126 (8): 89- 103
|
28 |
Liu Y , Du Y W , Li H , Liu F W , Wang Y J . Some second-order θ schemes combined with finite element method for nonlinear fractional cable equation. Numer Algorithms, 2019, 80 (2): 533- 555
|
29 |
Liu Y , Yu Z D , Li H , Liu F W , Wang J F . Time two-mesh algorithm combined with finite element method for time fractional water wave model. Int J Heat Mass Tran, 2018, 120 (5): 1132- 1145
|
30 |
Marion M , Xu J C . Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J Numer Anal, 1995, 32 (4): 1170- 1184
|
31 |
Novick-Cohen A . On the viscous Cahn-Hilliard equation. In: Ball J M, ed. Material Instabilities in Continuum Mechanics and Related Mathematical Problems. Oxford: Oxford Univ Press, 1988 329- 342
|
32 |
Scala R , Schimperna G . On the viscous Cahn-Hilliard equation with singular potential and inertial term. AIMS Math, 2016, 1 (1): 64- 76
|
33 |
Shang Y Q . A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations. J Comput Phys, 2013, 233 (1): 210- 226
|
34 |
Shen J , Yang X F . Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin Dyn Syst, 2010, 28 (4): 1669- 1691
|
35 |
Wang D X , Du Q Q , Zhang J W , Jia H E . A fast time two-mesh algorithm for Allen-Cahn equation. Bull Malays Math Sci Soc, 2019, 43 (3): 1- 25
|
36 |
Wang L , Yu H J . Convergence analysis of an unconditionally energy stable linear Crank-Nicolson scheme for the Cahn-Hilliard equation. 2018, 51 (1): 89- 114
|
37 |
Wang Y J , Liu Y , Li H , Wang J F . Finite element method combined with second-order time discrete scheme for nonlinear fractional cable equation. Eur Phys J Plus, 2016, 131 (3): 1- 16
|
38 |
Wise S M , Lowengrub J S , Frieboes H B , Cristini V . Three-dimensional multispecies nonlinear tumor growth—I: model and numerical method. J Theoret Biol, 2008, 253 (3): 524- 543
|
39 |
Xu J C . Two-grid discretization technique for linear and nonlinear PDEs. SIAM J Numer Anal, 1996, 33 (5): 1759- 1777
|
40 |
Yang X F , Zhao J , He X M . Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J Comput Appl Math, 2018, 343: 80- 97
|
41 |
Yin B L , Liu Y , Li H , He S . Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions. J Comput Phys, 2019, 379: 351- 372
|
42 |
Zaeem M A , Kadiri H E , Horstemeyer M F , Khafizov M , Utegulov Z . Effects of internal stresses and intermediate phases on the coarsening of coherent precipitates: A phase-field study. Curr Appl Phys, 2012, 12 (2): 570- 580
|
43 |
Zhang Z R , Qiao Z H . An adaptive time-stepping strategy for the Cahn-Hilliard equation. Commun Comput Phys, 2012, 11 (4): 1261- 1278
|
44 |
Zheng S , Milani A . Global attractors for singular perturbations of the Cahn-Hilliard equations. J Differential Equations, 2005, 209 (1): 101- 139
|
45 |
Zhou S W , Wang M Y . Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct Multidiscip Optim, 2007, 33 (2): 89- 111
|
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