June  2020, 28(2): 1077-1093. doi: 10.3934/era.2020059

High-order energy stable schemes of incommensurate phase-field crystal model

School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, Xiangtan 411105, China

* Corresponding author: Kai Jiang, Email: kaijiang@xtu.edu.cn

Received  April 2020 Revised  May 2020 Published  June 2020

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the $ L^2 $ gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.

Citation: Kai Jiang, Wei Si. High-order energy stable schemes of incommensurate phase-field crystal model. Electronic Research Archive, 2020, 28 (2) : 1077-1093. doi: 10.3934/era.2020059
References:
[1]

P. Bak, Phenomenological theory of icosahedral incommensurate ("quasiperiodic") order in Mn-Al alloys, Phys. Rev. Lett., 54 (1985), 1517-1519.  doi: 10.1103/PhysRevLett.54.1517.  Google Scholar

[2]

C. Corduneanu, Almost Periodic Functions, 2$^{nd}$ edition, Chelsea Publishing Company, New York, 1989. Google Scholar

[3]

H. Davenport and K. Mahler, Simultaneous Diophantine approximation, Duke Math. J., 13 (1946), 105-111.  doi: 10.1215/S0012-7094-46-01311-7.  Google Scholar

[4]

A. DuttL. Greengard and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math., 40 (2000), 241-266.  doi: 10.1023/A:1022338906936.  Google Scholar

[5]

R. Guo and Y. Xu, Semi-implicit spectral deferred correction method based on the invariant energy quadratization approach for phase field problems, Commun. Comput. Phys., 26 (2019), 87-113.  doi: 10.4208/cicp.OA-2018-0034.  Google Scholar

[6]

M. V. Jarić, Long-range icosahedral orientational order and quasicrystals., Phys. Rev. Lett., 55 (1985), 607-610.   Google Scholar

[7]

K. Jiang and W. Si, Stability of three-dimensional icosahedral quasicrystals in multi-component systems, Philos. Mag., 100 (2020), 84-109.  doi: 10.1080/14786435.2019.1671997.  Google Scholar

[8]

K. Jiang, J. Tong, P. Zhang and A.-C. Shi, Stability of two-dimensional soft quasicrystals in systems with two length scales, Phys. Rev. E, 92 (2015), 042159. Google Scholar

[9]

K. Jiang and P. Zhang, Numerical methods for quasicrystals, J. Comput. Phys., 256 (2014), 428-440.  doi: 10.1016/j.jcp.2013.08.034.  Google Scholar

[10]

K. Jiang, P. Zhang and A.-C. Shi, Stability of icosahedral quasicrystals in a simple model with two-length scales, J. Phys.: Condens. Matter, 29 (2017), 124003. doi: 10.1088/1361-648X/aa586b.  Google Scholar

[11]

X. Li and J. Shen, Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Advances in Computational Mathematics, arXiv: 1907.07462, 46 (2020), Article number: 48. doi: 10.1007/s10444-020-09789-9.  Google Scholar

[12]

R. Lifshitz and H. Diamant, Soft quasicrystals–Why are they stable?, Philos. Mag., 87 (2007), 3021-3030.  doi: 10.1080/14786430701358673.  Google Scholar

[13]

R. Lifshitz and D. M. Petrich, Theoretical model for Faraday waves with multiple-frequency forcing, Phys. Rev. Lett., 79 (1997), 1261-1264.  doi: 10.1103/PhysRevLett.79.1261.  Google Scholar

[14]

S. SavitzM. Babadi and R. Lifshitz, Multiple-scale structures: From Faraday waves to soft-matter quasicrystals, IUCrJ, 5 (2018), 247-268.  doi: 10.1107/S2052252518001161.  Google Scholar

[15]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[16]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[17]

S. M. WiseC. 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: 10.1137/080738143.  Google Scholar

[18]

X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316.  doi: 10.1016/j.jcp.2016.09.029.  Google Scholar

show all references

References:
[1]

P. Bak, Phenomenological theory of icosahedral incommensurate ("quasiperiodic") order in Mn-Al alloys, Phys. Rev. Lett., 54 (1985), 1517-1519.  doi: 10.1103/PhysRevLett.54.1517.  Google Scholar

[2]

C. Corduneanu, Almost Periodic Functions, 2$^{nd}$ edition, Chelsea Publishing Company, New York, 1989. Google Scholar

[3]

H. Davenport and K. Mahler, Simultaneous Diophantine approximation, Duke Math. J., 13 (1946), 105-111.  doi: 10.1215/S0012-7094-46-01311-7.  Google Scholar

[4]

A. DuttL. Greengard and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math., 40 (2000), 241-266.  doi: 10.1023/A:1022338906936.  Google Scholar

[5]

R. Guo and Y. Xu, Semi-implicit spectral deferred correction method based on the invariant energy quadratization approach for phase field problems, Commun. Comput. Phys., 26 (2019), 87-113.  doi: 10.4208/cicp.OA-2018-0034.  Google Scholar

[6]

M. V. Jarić, Long-range icosahedral orientational order and quasicrystals., Phys. Rev. Lett., 55 (1985), 607-610.   Google Scholar

[7]

K. Jiang and W. Si, Stability of three-dimensional icosahedral quasicrystals in multi-component systems, Philos. Mag., 100 (2020), 84-109.  doi: 10.1080/14786435.2019.1671997.  Google Scholar

[8]

K. Jiang, J. Tong, P. Zhang and A.-C. Shi, Stability of two-dimensional soft quasicrystals in systems with two length scales, Phys. Rev. E, 92 (2015), 042159. Google Scholar

[9]

K. Jiang and P. Zhang, Numerical methods for quasicrystals, J. Comput. Phys., 256 (2014), 428-440.  doi: 10.1016/j.jcp.2013.08.034.  Google Scholar

[10]

K. Jiang, P. Zhang and A.-C. Shi, Stability of icosahedral quasicrystals in a simple model with two-length scales, J. Phys.: Condens. Matter, 29 (2017), 124003. doi: 10.1088/1361-648X/aa586b.  Google Scholar

[11]

X. Li and J. Shen, Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Advances in Computational Mathematics, arXiv: 1907.07462, 46 (2020), Article number: 48. doi: 10.1007/s10444-020-09789-9.  Google Scholar

[12]

R. Lifshitz and H. Diamant, Soft quasicrystals–Why are they stable?, Philos. Mag., 87 (2007), 3021-3030.  doi: 10.1080/14786430701358673.  Google Scholar

[13]

R. Lifshitz and D. M. Petrich, Theoretical model for Faraday waves with multiple-frequency forcing, Phys. Rev. Lett., 79 (1997), 1261-1264.  doi: 10.1103/PhysRevLett.79.1261.  Google Scholar

[14]

S. SavitzM. Babadi and R. Lifshitz, Multiple-scale structures: From Faraday waves to soft-matter quasicrystals, IUCrJ, 5 (2018), 247-268.  doi: 10.1107/S2052252518001161.  Google Scholar

[15]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.  Google Scholar

[16]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.  Google Scholar

[17]

S. M. WiseC. 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: 10.1137/080738143.  Google Scholar

[18]

X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316.  doi: 10.1016/j.jcp.2016.09.029.  Google Scholar

Figure 1.  The spectral distribution and the corresponding real morphology of the initial value
Figure 2.  The time evolution of energy which is obtained by the SAV/CN scheme in the case of $ N_{T} = 256 $. The model parameters are $ q_{1} = 1 $, $ q_{2} = 2\cos(\pi/12) $, $ \tilde{\varepsilon} = -2 $ and $ \tilde{\alpha} = 2 $
Figure 3.  The morphologies of dynamic evolution in Fig. 2. Snapshots are taken at $ t = 50, 100, 150, 200 $, respectively
Figure 4.  The difference between the numerical energy values and the reference value $ E_{s} $. The numerical energy values are computed by the two methods: SAV/CN and SAV/CN+SDC. The reference energy value is obtained by the SAV/CN+SDC method in the case of $ N_{T} = 2048 $. The model parameters are $ q_{1} = 1 $, $ q_{2} = 2\cos(\pi/12) $, $ \tilde{\varepsilon} = -2 $ and $ \tilde{\alpha} = 2 $
Figure 5.  The real morphologies which describe the difference between the numerical solutions and the reference value at $ t = 0.4025 $. The reference solution is obtained by the SDC/CN+SDC method in the case of $ N_{T} = 2048 $. The model parameters are set as $ q_{1} = 1 $, $ q_{2} = 2\cos(\pi/12) $, $ \tilde{\varepsilon} = -2 $ and $ \tilde{\alpha} = 2 $
Figure 6.  The energy evolutions, initial values and convergence solutions of $ m $-length-scale DDQCs under the model parameters $ \tilde{\varepsilon} = -2 $, $ \tilde{\alpha} = 2 $. The scale parameters are set as $ q_{j} = s^{j-1} $, $ j = 1, \cdots, m $, $ s = 2\cos(\pi/12) $
Table 1.  Errors and convergence rates of the SAV/CN and SAV/CN+SDC schemes for the Allen-Cahn equation. The numerical solution of $ N_{T} = 2048 $ is regarded as the reference value
64 128 256 512
SAV/CN Error 4.75E-3 1.17E-3 2.91E-4 7.17E-5
Rate - 2.01 2.02 2.07
SAV/CN + SDC Error 1.16E-5 6.78E-7 4.04E-8 2.46E-9
Rate - 4.07 4.03 4.02
64 128 256 512
SAV/CN Error 4.75E-3 1.17E-3 2.91E-4 7.17E-5
Rate - 2.01 2.02 2.07
SAV/CN + SDC Error 1.16E-5 6.78E-7 4.04E-8 2.46E-9
Rate - 4.07 4.03 4.02
[1]

Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, , () : -. doi: 10.3934/era.2020089

[2]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033

[3]

Qiumei Huang, Xiaofeng Yang, Xiaoming He. Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2177-2192. doi: 10.3934/dcdsb.2018230

[4]

Tina Hartley, Thomas Wanner. A semi-implicit spectral method for stochastic nonlocal phase-field models. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 399-429. doi: 10.3934/dcds.2009.25.399

[5]

Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068

[6]

Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539

[7]

Marc Wolff, Stéphane Jaouen, Hervé Jourdren, Eric Sonnendrücker. High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 345-367. doi: 10.3934/dcdss.2012.5.345

[8]

José Luiz Boldrini, Gabriela Planas. A tridimensional phase-field model with convection for phase change of an alloy. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 429-450. doi: 10.3934/dcds.2005.13.429

[9]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117

[10]

Maciek D. Korzec, Hao Wu. Analysis and simulation for an isotropic phase-field model describing grain growth. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2227-2246. doi: 10.3934/dcdsb.2014.19.2227

[11]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[12]

Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016

[13]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems & Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[14]

Nobuyuki Kenmochi, Jürgen Sprekels. Phase-field systems with vectorial order parameters including diffusional hysteresis effects. Communications on Pure & Applied Analysis, 2002, 1 (4) : 495-511. doi: 10.3934/cpaa.2002.1.495

[15]

Maurizio Grasselli, Giulio Schimperna. Nonlocal phase-field systems with general potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5089-5106. doi: 10.3934/dcds.2013.33.5089

[16]

Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949

[17]

Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017

[18]

Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824

[19]

Alain Miranville, Elisabetta Rocca, Giulio Schimperna, Antonio Segatti. The Penrose-Fife phase-field model with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4259-4290. doi: 10.3934/dcds.2014.34.4259

[20]

Lela Dorel. Glucose level regulation via integral high-order sliding modes. Mathematical Biosciences & Engineering, 2011, 8 (2) : 549-560. doi: 10.3934/mbe.2011.8.549

2018 Impact Factor: 0.263

Article outline

Figures and Tables

[Back to Top]