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Numerical Algorithms
Erschienen in: Numerical Algorithms 1/2020

14.09.2019 | Original Paper

Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation

verfasst von: Zhengguang Liu, Xiaoli Li

Erschienen in: Numerical Algorithms | Ausgabe 1/2020

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Abstract

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be applied to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. We propose a series of efficient modified stabilized invariant energy quadratization approaches with unconditional energy stability for the phase-field crystal model. Firstly, we propose a more suitable positive preserving function strictly in square root and consider a modified invariant energy quadratization (MIEQ) approach. Secondly, a series of efficient and suitable functionals H(ϕ) in square root are considered and the modified stabilized invariant energy quadratization (MSIEQ) approaches are developed. We prove the unconditional energy stability and optimal error estimates for the semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, MIEQ, and MSIEQ approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.
Vorheriger Artikel An efficient method for computing the outer inverse through Gauss-Jordan elimination
Nächster Artikel Regions of convergence and dynamics of Schröder-like iteration formulae as applied to complex polynomial equations with multiple roots

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Literatur
1.
Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51, 2851–2873 (2013)MathSciNetMATH
2.
Bates, P.W., Brown, S., Han, J.: Numerical analysis for a nonlocal Allen-Cahn equation. Int. J. Numer. Anal. Mod 6, 33–49 (2009)MathSciNetMATH
3.
Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.M.: A linear energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 52, 546–562 (2012)MathSciNetMATH
4.
Chen, W., Wang, C., Wang, X., Wise, S.M.: A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection. J. Sci. Comput. 59, 574–601 (2014)MathSciNetMATH
5.
Chen, Y., Shen, J.: Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models. J. Comput. Phys. 308, 40–56 (2016)MathSciNetMATH
6.
Cheng, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation. J. Comput. Appl. Math. 362, 574–595 (2019)MathSciNetMATH
7.
Dehghan, M., Mohammadi, V.: The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods. Comput. Methods Appl. Mech. Eng. 298, 453–484 (2016)MathSciNetMATH
8.
Du, Q., Ju, L., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation. J. Comput. Phys. 363, 39–54 (2018)MathSciNetMATH
9.
Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. MRS Online Proceedings Library Archive, 529 (1998)
10.
Feng, W., Guan, Z., Lowengrub, J., Wang, C., Wise, S.M., Chen, Y.: A uniquely solvable, energy stable numerical scheme for the functionalized Cahn–Hilliard equation and its convergence analysis. J. Sci. Comput. 76, 1938–1967 (2018)MathSciNetMATH
11.
Feng, W., Wang, C., Wise, S.M., Zhang, Z.: A second-order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection. Numer. Methods Partial Differ. Equ. 34, 1975–2007 (2018)MathSciNetMATH
12.
Grasselli, M., Pierre, M.: Energy stable and convergent finite element schemes for the modified phase field crystal equation. ESAIM: Math. Modell. Numer. Anal. 50, 1523–1560 (2016)MathSciNetMATH
13.
He, Y., Liu, Y., Tang, T.: On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. 57, 616–628 (2007)MathSciNetMATH
14.
Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. 228, 5323–5339 (2009)MathSciNetMATH
15.
Lee, H.G., Kim, J.: A simple and efficient finite difference method for the phase-field crystal equation on curved surfaces. Comput. Methods Appl. Mech. Eng. 307, 32–43 (2016)MathSciNetMATH
16.
Li, H., Ju, L., Zhang, C., Peng, Q.: Unconditionally energy stable linear schemes for the diffuse interface model with Peng–Robinson equation of state. J. Sci. Comput. 75, 993–1015 (2018)MathSciNetMATH
17.
Li, Q., Mei, L., Yang, X., Li, Y.: Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation. Advances in Computational Mathematics, pp. 1–30 (2019)
18.
Li, Y., Kim, J.: An efficient and stable compact fourth-order finite difference scheme for the phase field crystal equation. Comput. Methods Appl. Mech. Eng. 319, 194–216 (2017)MathSciNetMATH
19.
Liu, F., Shen, J.: Stabilized semi-implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations. Math. Methods Appl. Sci. 38, 4564–4575 (2015)MathSciNetMATH
20.
Liu, Z., Li, X.: Efficient modified techniques of invariant energy quadratization approach for gradient flows. Appl. Math. Lett. 98, 206–214 (2019)MathSciNetMATH
21.
Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differ. Equ. 29, 584–618 (2013)MathSciNetMATH
22.
Praetorius, S., Voigt, A.: A Navier-Stokes phase-field crystal model for colloidal suspensions. J. Chem. Phys. 142, 154904 (2015)
23.
Qiao, Z., Sun, Z. -Z., Zhang, Z.: The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model. Numer. Methods Partial Differ. Equ. 28, 1893–1915 (2012)MathSciNetMATH
24.
Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50, 105–125 (2012)MathSciNetMATH
25.
Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)MathSciNetMATH
26.
Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discret. Cont. Dyn-A 28, 1669–1691 (2010)MathSciNetMATH
27.
Shin, J., Lee, H.G., Lee, J. -Y.: First and second order numerical methods based on a new convex splitting for phase-field crystal equation. J. Comput. Phys. 327, 519–542 (2016)MathSciNetMATH
28.
Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)MathSciNetMATH
29.
Weng, Z., Zhai, S., Feng, X.: A fourier spectral method for fractional-in-space Cahn–Hilliard equation. Appl. Math. Model. 42, 462–477 (2017)MathSciNetMATH
30.
Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47, 2269–2288 (2009)MathSciNetMATH
31.
Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys. 23, 572–602 (2018)MathSciNet
32.
Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294–316 (2016)MathSciNetMATH
33.
Yang, X., Han, D.: Linearly first-and second-order, unconditionally energy stable schemes for the phase field crystal model. J. Comput. Phys. 330, 1116–1134 (2017)MathSciNetMATH
34.
Yang, X., Zhang, G.: Numerical approximations of the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential using the Invariant Energy Quadratization approach, arXiv:1712.​02760 (2017)
35.
Zhang, L., Zhou, Z.: Spectral galerkin approximation of optimal control problem governed by Riesz fractional differential equation. Appl. Numer. Math. 143, 247–262 (2019)MathSciNetMATH
36.
Zhao, J., Wang, Q., Yang, X.: Numerical approximations to a new phase field model for two phase flows of complex fluids. Comput. Methods Appl. Mech. Eng. 310, 77–97 (2016)MathSciNetMATH
37.
Zhou, Z., Chen, F., Chen, H.: Convergence analysis for H1-Galerkin mixed finite element approximation of one nonlinear integro-differential model. Appl. Math. Comput. 220, 783–791 (2013)MathSciNetMATH
38.
Zhou, Z., Yu, X., Yan, N.: Local discontinuous Galerkin approximation of convection-dominated diffusion optimal control problems with control constraints. Numer. Methods Partial Differ. Equ. 30, 339–360 (2014)MathSciNetMATH
39.
Zhou, Z., Zhang, C.: Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation. Numer. Algorithm. 79, 437–455 (2018)MathSciNetMATH
40.
Zhu, J., Chen, L.-Q., Shen, J., Tikare, V.: Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. Phys. Rev. E. 60, 3564 (1999)
Metadaten
Titel
Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation
verfasst von
Zhengguang Liu
Xiaoli Li
Publikationsdatum
14.09.2019
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 1/2020
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00804-9

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