Volume 26, Issue 5
An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Commun. Comput. Phys., 26 (2019), pp. 1335-1364.

Published online: 2019-08

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• Abstract

In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower than that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the $ℓ$(0,$T$;$ℓ$2)∩$ℓ$2(0,$T$;$H_N^3$) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

• Keywords

Square phase field crystal equation, Fourier pseudo-spectral approximation, second order BDF stencil, energy stability, optimal rate convergence analysis, preconditioned steepest descent iteration.

• AMS Subject Headings

35K30, 35K55, 65K10, 65M12, 65M70

zhengkelong@swust.edu.cn (Kelong Cheng)

swise1@utk.edu (Steven M. Wise)

• BibTex
• RIS
• TXT
@Article{CiCP-26-1335, author = {Cheng , Kelong and Wang , Cheng and M. Wise , Steven }, title = {An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {5}, pages = {1335--1364}, abstract = {

In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower than that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the $ℓ$(0,$T$;$ℓ$2)∩$ℓ$2(0,$T$;$H_N^3$) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2019.js60.10}, url = {http://global-sci.org/intro/article_detail/cicp/13267.html} }
TY - JOUR T1 - An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation AU - Cheng , Kelong AU - Wang , Cheng AU - M. Wise , Steven JO - Communications in Computational Physics VL - 5 SP - 1335 EP - 1364 PY - 2019 DA - 2019/08 SN - 26 DO - http://dor.org/10.4208/cicp.2019.js60.10 UR - https://global-sci.org/intro/article_detail/cicp/13267.html KW - Square phase field crystal equation, Fourier pseudo-spectral approximation, second order BDF stencil, energy stability, optimal rate convergence analysis, preconditioned steepest descent iteration. AB -

In this paper we propose and analyze an energy stable numerical scheme for the square phase field crystal (SPFC) equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. In particular, a modification of the free energy potential to the standard phase field crystal model leads to a composition of the 4-Laplacian and the regular Laplacian operators. To overcome the difficulties associated with this highly nonlinear operator, we design numerical algorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure is respected, and summation-by-parts formulae enable us to study the discrete energy stability for such a high-order spatial discretization. In the temporal approximation, a second order BDF stencil is applied, combined with an appropriate extrapolation for the concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to the artificial linear diffusion coming at an order lower than that of surface diffusion term. Such a choice leads to reduced numerical dissipation. At a theoretical level, the unique solvability, energy stability are established, and an optimal rate convergence analysis is derived in the $ℓ$(0,$T$;$ℓ$2)∩$ℓ$2(0,$T$;$H_N^3$) norm. In the numerical implementation, the preconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and a geometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

Kelong Cheng, Cheng Wang & Steven M. Wise. (2019). An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation. Communications in Computational Physics. 26 (5). 1335-1364. doi:10.4208/cicp.2019.js60.10
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