Low regularity integrators for semilinear parabolic equations with maximum bound principles
Low regularity integrators for semilinear parabolic equations with maximum bound principles
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DOI：
10.1007/s10543-023-00946-2
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发表时间：
2022-11
影响因子：
1.5
通讯作者：
Cao-Kha Doan;Thi-Thao-Phuong Hoang;L. Ju;Katharina Schratz
中科院分区：
文献类型：
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作者：
Cao-Kha Doan;Thi-Thao-Phuong Hoang;L. Ju;Katharina Schratz
研究了一类Allen-Cahn型半线性抛物型方程的条件保结构低正则性时间积分方法。这类方程的重要性质包括最大边界原理（MBP）和能量耗散定律;对于前者，这意味着解的绝对值始终由适当的初始和边界条件所施加的常数点态有界。首先采用中心差分法对模型方程进行空间离散，然后利用Duhamel公式迭代构造一阶和二阶低正则性积分器（LRI）对半离散系统进行时间离散。所提出的LRI格式被证明是保持MBP和能量稳定的离散意义。此外，他们的时间误差估计也成功地获得了低的正则性要求下，半离散问题的精确解只假设是连续的时间。数值结果表明，与经典的指数时间差分格式相比，所提出的LRI格式具有更高的精度和更好的收敛速度，特别是当界面参数趋于零时.
This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen–Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.