Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation
Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation
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DOI：
10.1007/s10444-020-09782-2
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发表时间：
2019-06
影响因子：
1.7
通讯作者：
Bingquan Ji;Hong-lin Liao;Lu-ming Zhang
中科院分区：
文献类型：
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作者：
Bingquan Ji;Hong-lin Liao;Lu-ming Zhang
针对具有Caputo导数的时间分数Allen-Cahn方程，提出了两种快速的L1时间推进方法，包括向后Euler格式和稳定化半隐式格式。在初始时间附近对时间网格进行细化，以解决解的固有初始奇异性，并在方法中加入不等时间步长，以便在长时间模拟中使用自适应时间步长策略。结果表明，采用快速L1公式的格式保持了离散极大值原理。利用离散分数Grönwall不等式和全局相合性分析，建立了反映解的时间正则性的尖锐误差估计。数值实验表明了我们方法的有效性，并验证了我们的分析。
Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo’s derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time steps are always incorporated into our approaches so that a adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.