论文标题
无条件的能量耗散，自适应IMEX BDF2方案及其在广义SAV方法上的Cahn-Hilliard方程的误差估计值
A unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn-Hilliard equation on generalized SAV approach
论文作者
论文摘要
通过与空间中的傅立叶光谱方法结合，对Cahn-Hilliard方程的广义SAV方法进行了自适应隐式解释（IMEX）BDF2方案。事实证明，修改的能量耗散法在离散级别无条件保存。在温和的比率限制下，即\ ass {1}：$ 0 <r_k：=τ_k/τ_{k-1} <r _ {\ max} \大约4.8645 $，我们在$ h^1 $ norm中建立了严格的误差估计，并实现最佳的二级准确度。证明涉及离散正交卷积（DOC）内核和不平等变焦的工具。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骤求解一个线性系统恒定系数。在我们的分析中，第一步的第一次一致性BDF1并未使订单减少$ h^1 $ -NORM。在周期性边界条件下，数值解的$ h^1 $可以在没有任何限制的情况下得出（例如初始数据的零均值）。最后，提供了数值示例，以验证我们的理论分析和算法效率。
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., \Ass{1}: $0<r_k:=τ_k/τ_{k-1}< r_{\max}\approx 4.8645$, 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 the 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.
