A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation

Abstract

In this paper we propose and analyze a second order accurate (in time) numerical scheme for the square phase field crystal equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. Its primary difference with the standard phase field crystal model is an introduction of the 4-Laplacian term in the free energy potential, which in turn leads to a much higher degree of nonlinearity. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a second order backward differentiation formula is applied in the temporal stencil. Meanwhile, a direct application of the SAV method faces certain difficulties, due to the involvement of the 4-Laplacian term, combined with a derivation of the lower bound of the nonlinear energy functional. In the proposed numerical method, an appropriate decomposition for the physical energy functional is formulated, so that the nonlinear energy part has a well-established global lower bound, and the rest terms lead to constant-coefficient diffusion terms with positive eigenvalues. In turn, the numerical scheme could be very efficiently implemented by constant-coefficient Poisson-like type solvers (via FFT), and energy stability is established by introducing an auxiliary variable, and an optimal rate convergence analysis is provided for the proposed SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

Appendix

1.1 Proof of Proposition 2.4

Due to the periodic boundary condition for f and its cell-centered representation, it has a corresponding discrete Fourier transformation, as the form given by (2.3):

$$\begin{aligned} f_{i,j,k} = \sum _{\ell ,m,n=-K}^{K} {\hat{f}}_{\ell ,m,n}^N \exp \left( 2 \pi \mathrm{i} ( \ell x_i + m y_j + n z_k ) \right) . \end{aligned}$$

(A.1)

Then we make its extension to a continuous function:

$$\begin{aligned} f_N (x,y,z) = \sum ^{K}_{\ell ,m,n=-K} {\hat{f}}^N_{\ell ,m,n} \exp \left( 2 \pi \mathrm{i} ( \ell x + m y + n z ) \right) . \end{aligned}$$

(A.2)

We denote a discrete grid function, $g := \mathcal{D}_x f$, at a point-wise level. Since f corresponds to $f_N \in \mathcal{B}^K$ (the space of trigonometric polynomials of degree at most K), an application of Parseval identity implies that

$$\begin{aligned} \begin{aligned}&\Vert \nabla _N \Delta _N f \Vert _2^2 = \Vert \nabla \Delta f_N \Vert ^2 = \sum ^{K}_{\ell ,m,n=-K} \lambda _{\ell , m, n}^6 | {\hat{f}}^N_{\ell ,m,n} |^2 , \\&\quad \Vert \Delta _N^3 f \Vert _2^2 = \Vert \Delta ^3 f_N \Vert ^2 = \sum ^{K}_{\ell ,m,n=-K} \lambda _{\ell , m, n}^{12} | {\hat{f}}^N_{\ell ,m,n} |^2 , \end{aligned} \end{aligned}$$

(A.3)

with $\lambda _{\ell , m, n}$ introduced in (2.13). Meanwhile, the elliptic regularity for the continuous function $f_N$ indicates that

$$\begin{aligned} \Vert \nabla \Delta f_N \Vert \le {\hat{C}}_0 \Vert \Delta ^3 f_N \Vert , \quad \text{ for } \text{ some } {\hat{C}}_0\text { only dependent on }\Omega . \end{aligned}$$

(A.4)

Finally, the discrete elliptic regularity inequality (2.24) is a direct combination of  (A.3) and (A.4). This completes the proof of Proposition 2.4.