论文标题
能量稳定和最大绑定的保留方案，具有可变的时间步长的时间分数allen-cahn方程
An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
论文作者
论文摘要
在这项工作中，我们提出了一个曲柄 - 尼科尔森型方案，该方案在时间分数方程中具有可变步骤。所提出的方案被证明是无条件稳定的（在各种能量意义上），并且是最大结合的。有趣的是，本文获得的离散能量稳定性结果可以恢复经典的能量耗散法，当分数订单$α\ rightarrow 1. $即，我们的方案可以渐近地保留$α\ rightarrow 1 $限制的能量消散法。这似乎是可变的时间稳定方案的第一项工作，它可以保留能量稳定性和最大结合原理。 我们的曲柄尼科尔森计划是建立在与Riemann-Liouville衍生品相关的重新制定问题上的。作为产品，我们借助一类离散的正交卷积内核，在Riemann-Liouville衍生物的L1型公式和新的L1型公式之间建立了可逆的转换。这是第一次在两个离散的分数衍生物之间建立这样的\ textit {iNCETE}转换。最终，我们提出了几个数值示例，并具有自适应的时间步变策略，以显示拟议方案的有效性。
In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $α\rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $α\rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.
