@Article{CiCP-26-1335,
author = {Cheng , KelongWang , 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 = {<p style="text-align: justify;">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 $ℓ$<sup>∞</sup>(0,$T$;$ℓ$<sup>2</sup>)∩$ℓ$<sup>2</sup>(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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.2019.js60.10},
url = {http://global-sci.org/intro/article_detail/cicp/13267.html}
}