求解Allen-Cahn方程的两种高效数值格式
Two Efficient Numerical Schemes for the Allen-Cahn Equation
摘要: 基于算子分裂思想，本文提出了求解Allen-Cahn方程的两种高效算子分裂格式。首先将此方程分裂为线性项与非线性项两个部分：对线性部分，通过二阶中心差分法与四阶紧致差分法分别进行数值计算，并利用傅里叶方法给出了稳定性的分析；对非线性部分进行精确求解。最后，通过数值算例比较两种算子分裂格式的数值解差异以及运行效率，而且验证了两种格式满足能量递减规律。
Abstract: Based on the idea of operator splitting, this paper proposes two efficient operator splitting schemes for the Allen-Cahn equation. The original equation is divided into linear and nonlinear parts. The linear part is approximated numerically by two different schemes: the second-order center difference scheme and the fourth-order compact difference scheme. The stability analysis of both schemes is discussed according to a Fourier stability analysis. The nonlinear part is solved accurately. Numerical comparisons are carried out to verify the accuracy and efficiency of the proposed methods, and to verify the given schemes satisfied the law of decreasing energy.
文章引用：郑楠, 翟术英, 翁智峰. 求解Allen-Cahn方程的两种高效数值格式[J]. 应用数学进展, 2017, 6(3): 283-295. https://doi.org/10.12677/AAM.2017.63034

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