In this paper, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function $\phi^{n}$ by setting the positive constant $C_{0}$ large enough such that $C_{0}\geq 1/\Dt.$ The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio $\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. Furthermore, a rigorous error estimate of the second order accuracy of $\phi^{n}$ is derived for the proposed scheme on the nonuniform mesh by using the uniform $H^{2}$ bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and demonstrate the efficiency of the fully discrete adaptive BDF2 scheme.

翻译：本文针对相场晶体方程提出并分析了一种采用变时间步长的线性全离散二阶格式。具体而言，我们基于二阶后向差分公式（BDF2）构建了一种线性自适应时间步进格式，并采用傅里叶谱方法进行空间离散。利用标量辅助变量方法处理非线性项，其中仅采用一阶方法近似辅助变量。这一处理对于所提出的自适应BDF2格式无条件能量稳定性的推导至关重要。然而，我们首次发现，通过将正常数$C_{0}$设定为足够大（满足$C_{0}\geq 1/\Dt$），该策略不会影响未知相函数$\phi^{n}$的二阶精度。在相邻时间步长比$\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$的温和约束下，建立了自适应BDF2格式的能量稳定性。此外，利用数值解的一致$H^{2}$有界性，在非均匀网格上推导了$\phi^{n}$二阶精度的严格误差估计。最后，通过数值实验验证了理论结果，并证明了全离散自适应BDF2格式的高效性。