# American Institute of Mathematical Sciences

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. Dutt, L. 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. Savitz, M. 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. Shen, J. 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. 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: 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

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##### 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. Dutt, L. 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. Savitz, M. 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. Shen, J. 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. 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: 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
The spectral distribution and the corresponding real morphology of the initial value
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 $\tilde{\alpha} = 2$
. Snapshots are taken at $t = 50, 100, 150, 200$, respectively">Figure 3.  The morphologies of dynamic evolution in Fig. 2. Snapshots are taken at $t = 50, 100, 150, 200$, respectively
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 $\tilde{\alpha} = 2$
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 $\tilde{\alpha} = 2$
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 $q_{j} = s^{j-1}$, $j = 1, \cdots, m$, $s = 2\cos(\pi/12)$
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 0.00475 0.00117 0.000291 7.17E-5 Rate - 2.01 2.02 2.07 SAV/CN + SDC Error 1.16e-05 6.78e-07 4.04e-08 2.46E-9 Rate - 4.07 4.03 4.02
 64 128 256 512 SAV/CN Error 0.00475 0.00117 0.000291 7.17E-5 Rate - 2.01 2.02 2.07 SAV/CN + SDC Error 1.16e-05 6.78e-07 4.04e-08 2.46E-9 Rate - 4.07 4.03 4.02
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