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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DOI：
10.4208/cicp.2019.js60.10
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发表时间：
2019-06
期刊：
影响因子：
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通讯作者：
Kelong Cheng;Cheng Wang;S. Wise
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文献类型：
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作者：
Kelong Cheng;Cheng Wang;S. Wise
在本文中，我们提出并分析了一个能量稳定的数值方相场晶体（SPFC）方程，梯度流模拟晶体动力学在原子尺度上的空间，但在时间上的扩散尺度。特别是，修改的自由能势的标准相场晶体模型导致的4-拉普拉斯算子和定期拉普拉斯算子的组合物。为了克服与这种高度非线性算子相关的困难，我们设计了基于单个能量项结构的数值算法。傅立叶伪谱近似是在空间中，在这样一种方式，能量结构的尊重，和求和的部分公式使我们能够研究这样一个高阶空间离散化的离散能量稳定性。在时间近似中，应用二阶BDF模板，结合凹扩散项的适当外推。为了保证能量的稳定性，增加了二阶人工Douglas-Dupont型正则化项，并通过分析得到人工线性扩散项的阶数低于表面扩散项的阶数.这样的选择导致减少的数值耗散。在理论水平上，建立了唯一的可解性，能量稳定性，并在$\ell^\infty（0，T; \ell^2）\cap \ell^2（0，T; H_N^3）$范数下导出了最优速率收敛分析。在数值实现中，采用预条件最速下降（PSD）迭代求解高度非线性的4-Laplacian项和标准Laplacian项的合成，并保证了这种迭代的几何收敛性.最后，给出了一些数值实验，证实了所提出的方案的鲁棒性和精度。
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 that 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 $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_N^3)$ 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.