论文标题
时间分数Cahn-Hilliard模型的可变步长L1型方案的能量稳定性
Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model
论文作者
论文摘要
离散时间分数衍生物的积极确定性对于时间段相位场模型的数值稳定性（在能量意义上）至关重要。提出了一种新型技术来估计由非均匀L1，基于半网格的L1和分数Caputo衍生物的时间平均L1公式产生的离散卷积内核的最低特征值。主要的离散工具是离散的正交卷积内核和离散的互补卷积内核。然后以分离时间分数的Cahn-Hilliard模型建立了某些以离散级别的分散式L1型方法的分离级别耗散法律。这些方法被证明是在分数限制限制$α\ rightArrow1 $中均不兼容的。提供了数值示例，以及提供自适应的时间步骤程序，以证明所提出的方法的有效性。
The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo's derivative. The main discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Certain variational energy dissipation laws at discrete levels of the variable-step L1-type methods are then established for time-fractional Cahn-Hilliard model.They are shown to be asymptotically compatible, in the fractional order limit $α\rightarrow1$, with the associated energy dissipation law for the classical Cahn-Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.
