Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation
Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation
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DOI：
10.1002/num.23017
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发表时间：
2023-03
影响因子：
3.9
通讯作者：
Cao-Kha Doan;Thi-Thao-Phuong Hoang;L. Ju
中科院分区：
文献类型：
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作者：
Cao-Kha Doan;Thi-Thao-Phuong Hoang;L. Ju
艾伦-卡恩方程满足最大界原理，即在适当的初始和/或边界条件下，它的解在任何时候都一致有界于一个正常数。最近已经证明，由低正则性积分器（LRI）产生的时间离散解同样在无穷范数下有界;然而，仍然缺乏相应的完全离散误差分析。研究了基于时间上的两个一阶LRI和空间上的中心有限差分方法的艾伦-卡恩方程全离散数值解的收敛性.利用完全离散系统的一些基本性质和Duhamel原理，证明了当精确解仅在时间上连续时，数值解在时间和空间上的最优误差估计.数值结果证实了这种误差估计，并表明，所提出的LRI计划获得的解决方案是更准确的比经典的指数时间差分（ETD）计划的相同的顺序。
The Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.