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http://dx.doi.org/10.12941/jksiam.2019.23.001

A CONSTRAINED CONVEX SPLITTING SCHEME FOR THE VECTOR-VALUED CAHN-HILLIARD EQUATION  

LEE, HYUN GEUN (DEPARTMENT OF MATHEMATICS, KWANGWOON UNIVERSITY)
LEE, JUNE-YUB (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
SHIN, JAEMIN (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.23, no.1, 2019 , pp. 1-18 More about this Journal
Abstract
In contrast to the well-developed convex splitting schemes for gradient flows of two-component system, there were few efforts on applying the convex splitting idea to gradient flows of multi-component system, such as the vector-valued Cahn-Hilliard (vCH) equation. In the case of the vCH equation, one need to consider not only the convex splitting idea but also a specific method to manage the partition of unity constraint to design an unconditionally energy stable scheme. In this paper, we propose a constrained Convex Splitting (cCS) scheme for the vCH equation, which is based on a convex splitting of the energy functional for the vCH equation under the constraint. We show analytically that the cCS scheme is mass conserving and unconditionally uniquely solvable. And it satisfies the constraint at the next time level for any time step thus is unconditionally energy stable. Numerical experiments are presented demonstrating the accuracy, energy stability, and efficiency of the proposed cCS scheme.
Keywords
Vector-valued Cahn-Hilliard equation; Constrained convex splitting; Unconditional unique solvability; Unconditional energy stability;
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1 U.M. Ascher, S.J. Ruuth, and R.J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), 151-167.   DOI
2 V.E. Badalassi, H.D. Ceniceros, and S. Banerjee, Computation of multiphase systems with phase field models, J. Comput. Phys. 190 (2003), 371-397.   DOI
3 J.W. Barrett and J.F. Blowey, An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numer. Math. 72 (1995), 1-20.   DOI
4 J.W. Barrett and J.F. Blowey, An error bound for the finite element approximation of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal. 16 (1996), 257-287.   DOI
5 J.F. Blowey, M.I.M. Copetti, and C.M. Elliott, Numerical analysis of a model for phase separation of a multicomponent alloy, IMA J. Numer. Anal. 16 (1996), 111-139.   DOI
6 F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model, ESAIM: M2AN 40 (2006), 653-687.   DOI
7 F. Boyer and S. Minjeaud, Hierarchy of consistent n-component Cahn-Hilliard systems, Math. Models Meth. Appl. Sci. 24 (2014), 2885-2928.   DOI
8 J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267.   DOI
9 L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32 (2000), 113-140.   DOI
10 J.P. Desi, H.H. Edrees, J.J. Price, E. Sander, and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM J. Appl. Dyn. Syst. 10 (2011), 707-743.   DOI
11 S. Dong, An efficient algorithm for incompressible N-phase flows, J. Comput. Phys. 276 (2014), 691-728.   DOI
12 C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy, IMA Preprint Series 887, 1991.
13 C.M. Elliott and A.M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (1993), 1622-1663.   DOI
14 D.J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Proc. 529 (1998), 39-46.   DOI
15 D. de Fontaine, A computer simulation of the evolution of coherent composition variations in solid solutions, Ph.D. Thesis, Northwestern University, 1967.
16 H. Garcke, B. Nestler, and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits, Phys. D 115 (1998), 87-108.   DOI
17 Z. Guan, C. Wang, and S.M. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn- Hilliard equation, Numer. Math. 128 (2014), 377-406.   DOI
18 D. Jeong and J. Kim, A practical numerical scheme for the ternary Cahn-Hilliard system with a logarithm free energy, Phys. A 442 (2016), 510-522.   DOI
19 D. Jeong and J. Kim, Practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy, Math. Meth. Appl. Sci. 40 (2017), 1734-1745.   DOI
20 J. Kim, K. Kang, and J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard systems, Comm. Math. Sci. 2 (2004), 53-77.   DOI
21 H.G. Lee, J.-W. Choi, and J. Kim, A practically unconditionally gradient stable scheme for the N-component Cahn-Hilliard system, Phys. A 391 (2012), 1009-1019.   DOI
22 J.E. Morral and J.W. Cahn, Spinodal decomposition in ternary systems, Acta Metall. 19 (1971), 1037-1045.   DOI
23 H.G. Lee and J. Kim, A second-order accurate non-linear difference scheme for the N-component Cahn- Hilliard system, Phys. A 387 (2008), 4787-4799.   DOI
24 H.G. Lee and J. Kim, An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations, Comput. Phys. Commun. 183 (2012), 2107-2115.   DOI
25 H.G. Lee, J. Shin, and J.-Y. Lee, First- and second-order energy stable methods for the modified phase field crystal equation, Comput. Methods Appl. Mech. Engrg. 321 (2017), 1-17.   DOI
26 B. Nestler and A.A. Wheeler, A multi-phase-field model of eutectic and peritectic alloys: numerical simulation of growth structures, Phys. D 138 (2000), 114-133.   DOI
27 E. Oudet, Approximation of partitions of least perimeter by $\Gamma$-convergence: around Kelvin's conjecture, Exp. Math. 20 (2011), 260-270.   DOI
28 J. Shin, H.G. Lee, and J.-Y. Lee, First and second order numerical methods based on a new convex splitting for phase-field crystal equation, J. Comput. Phys. 327 (2016), 519-542.   DOI
29 J. Shin, H.G. Lee, and J.-Y. Lee, Convex Splitting Runge-Kutta methods for phase-field models, Comput. Math. Appl. 73 (2017), 2388-2403.   DOI
30 J. Shin, H.G. Lee, and J.-Y. Lee, Unconditionally stable methods for gradient flow using Convex Splitting Runge-Kutta scheme, J. Comput. Phys. 347 (2017), 367-381.   DOI
31 R. Tavakoli, Computationally efficient approach for the minimization of volume constrained vector-valued Ginzburg-Landau energy functional, J. Comput. Phys. 295 (2015), 355-378.   DOI
32 R. Tavakoli, Unconditionally energy stable time stepping scheme for Cahn-Morral equation: Application to multi-component spinodal decomposition and optimal space tiling, J. Comput. Phys. 304 (2016), 441-464.   DOI
33 G.I. Toth, T. Pusztai, and L. Granasy, Consistent multiphase-field theory for interface driven multidomain dynamics, Phys. Rev. B 92 (2015), 184105.   DOI
34 G.I. Toth, M. Zarifi, and B. Kvamme, Phase-field theory of multicomponent incompressible Cahn-Hilliard liquids, Phys. Rev. E 93 (2016), 013126.   DOI
35 L. Vanherpe, F. Wendler, B. Nestler, and S. Vandewalle, A multigrid solver for phase field simulation of microstructure evolution, Math. Comput. Simul. 80 (2010), 1438-1448.   DOI
36 C.Wang, X.Wang, and S.M.Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Cont. Dyn. S. 28 (2010), 405-423.   DOI
37 C.Wang and S.M.Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal. 49 (2011), 945-969.   DOI
38 S.M. Wise, C. Wang, and J.S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), 2269-2288.   DOI