An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=\tau _{k}/\tau _{k-1}< r_{\max }\approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.

中文翻译：

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广义 SAV 方法上的无条件能量耗散、自适应 IMEX BDF2 方案及其对 Cahn-Hilliard 方程的误差估计


结合空间傅里叶谱方法，研究了 Cahn-Hilliard 方程的广义 SAV 方法的自适应隐式-显式 (IMEX) BDF2 方案。证明了修正后的能量耗散定律在离散水平上无条件保持。在温和的比率限制下，即 A1: $0<r_{k}:=\tau _{k}/\tau _{k-1}< r_{\max }\approx 4.8645$，我们建立了严格的误差以 $H^{1}$-范数进行估计并及时实现最佳二阶精度。证明涉及离散正交卷积（DOC）核和不等式缩放工具。值得注意的是，所提出的自适应时间步长方案仅需要在每个时间步长求解一个具有常系数的线性系统。在我们的分析中，第一步的第一一致 BDF1 不会带来 $H^{1}$-范数的阶数减少。可以在没有任何限制（例如初始数据的零均值）的情况下推导出周期性边界条件下数值解的$H^{1}$界。最后，提供数值例子来验证我们的理论分析和算法的效率。