On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation
On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation
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DOI：
10.1137/19m1289157
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发表时间：
2020-01
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影响因子：
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通讯作者：
Hong-lin Liao;T. Tang;Tao Zhou
中科院分区：
文献类型：
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作者：
Hong-lin Liao;T. Tang;Tao Zhou
在这项工作中，我们研究了Allen-Cahn方程的非均匀网格两步向后微分公式(BDF2)。我们证明了非均匀BDF_2格式在时间步长比约束$r_k：=\tau_k/\tau_{k-1}<；(3+\sqrt{17})/2\approx3.561.$下是能量稳定的。此外，通过发展一种新的核重组和互补技术，我们首次证明了时间步长比约束$r_k<；1+\Sqrt{2}\约2.414美元和一个实际的时间步长约束下的BDF_2格式的离散极大值原理。给出了最大范数下的二阶收敛速度。数值实验结果支持了理论研究结果。
In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.