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An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation
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Year:    2019

Author:    Kelong Cheng, Cheng Wang, Steven M. Wise

Communications in Computational Physics, Vol. 26 (2019), Iss. 5 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.2019.js60.10

Communications in Computational Physics, Vol. 26 (2019), Iss. 5 : pp. 1335–1364

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

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

Author Details

Kelong Cheng

Cheng Wang

Steven M. Wise

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