A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960\le \tau _{k}/\tau _{k-1}<r^{*}(\alpha )$, where $\tau _{k}$ is the $k$th time-step size and $r^{*}(\alpha )\ge 4.660$ for $\alpha \in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.

中文翻译：

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时间分数 Cahn-Hilliard 模型的变步长分数 BDF2 格式的渐近兼容能量


通过局部-非局部分裂技术构造了逼近 $\alpha \in (0,1)$ 阶 Caputo 分数阶导数的变步长分数 BDF2 公式的离散梯度结构，即将分数 BDF2 公式进行分裂分为类似于一阶导数的两步向后微分公式 (BDF2) 的局部部分和类似于 Caputo 导数的 L1 型公式的非局部部分。然后，在弱步长比约束$0.3960\le \tau _{k}/\tau _{k- 下，为时间分数 Cahn-Hilliard 模型建立了变步长分数 BDF2 隐式格式的局部离散能量耗散律。 1}<r^{*}(\alpha )$，其中 $\tau _{k}$ 是第 $k$ 个时间步长，$r^{*}(\alpha )\ge 4.660$ 为 $ \alpha \in (0,1)$。目前的结果为 [SINUM, 57: 218-237, Remark 6] 中的开放问题提供了实际的答案，并显着放松了严格的步长比限制 [Math.比较，90：19–40，定理 3.2]。更有趣的是，离散能量和相应的能量耗散定律分别与经典 Cahn-Hilliard 方程的变步长 BDF2 方法的相关离散能量和能量耗散定律渐近兼容。据我们所知，这种类型的能量耗散律是首次为Caputo导数的变步长L2型公式建立。提供了自适应步进程序的数值示例来证明我们提出的方法的准确性和有效性。