Maximum norm error analysis of an unconditionally stable semi-implicit scheme for multi-dimensional Allen–Cahn equations
Maximum norm error analysis of an unconditionally stable semi-implicit scheme for multi-dimensional Allen–Cahn equations
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多维 Allen-Cahn 方程无条件稳定半隐式格式的最大范数误差分析
DOI：
10.1002/num.22333
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发表时间：
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期刊：
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通讯作者：
Kejia Pan
中科院分区：
文献类型：
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作者：
Dongdong He;Kejia Pan
本文提出了一种求解多维Allen-Cahn方程的线性化有限差分格式。该方案采用改进的跃蛙格式进行时间离散，对非线性项采用半隐式处理，在空间上采用中心差分格式进行离散。该方法满足离散能量衰减特性，具有无条件稳定性。此外，进行了严格的最大范数误差分析，表明该方法在时间和空间变量上都是二阶准确的。最后，给出了二维和三维问题的数值试验来证实我们的理论发现。
In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.