An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
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DOI：
10.1137/20m1384105
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发表时间：
2020-12
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影响因子：
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通讯作者：
Hong-lin Liao;T. Tang;Tao Zhou
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文献类型：
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作者：
Hong-lin Liao;T. Tang;Tao Zhou
在这项工作中，我们针对时间分数 Allen-Cahn 方程提出了一种具有可变步长的 Crank-Nicolson 型方案。所提出的方案被证明是无条件稳定的（在变分能量意义上），并且是最大界限保持的。有趣的是，本文得到的离散能量稳定性结果可以在分数阶 $\alpha \rightarrow 1.$ 时恢复经典的能量耗散定律，即我们的方案可以在 $\alpha \rightarrow 1$ 极限下渐近地保留能量耗散定律。这似乎是第一个关于可变时间步长方案的工作，该方案可以保持能量稳定性和最大界限原理。我们的克兰克-尼科尔森方案是建立在与黎曼-刘维尔导数相关的重新表述的问题之上的。作为副产品，我们借助一类离散正交卷积核，在黎曼-刘维尔导数的 L1 型公式和 Caputo 导数的新 L1 型公式之间建立了可逆变换。这是第一次在两个离散分数导数之间建立这样的 \textit{离散} 变换。最后，我们提出了几个具有自适应时间步进策略的数值示例，以展示所提出方案的有效性。
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 $\alpha \rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \rightarrow 1$ 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.