论文标题
半线性抛物线方程的低规律性集成符，具有最大界限原理
Low regularity integrators for semilinear parabolic equations with maximum bound principles
论文作者
论文摘要
本文与Allen-Cahn类型的一类半线性抛物线方程相关的有条件具有结构性的，低规律性的时间整合方法。此类方程的重要特性包括最大结合原理（MBP）和能量耗散定律；对于前者而言，这意味着解决方案的绝对值一直是通过适当的初始和边界条件施加的常数。首先通过中央有限差在空间中离散的模型方程，然后使用Duhamel的公式，一阶和二阶低规则性集成符（LRIS）进行迭代，以用于半污垢系统的时间离散化。事实证明，所提出的LRI方案可以在离散意义上保留MBP和能量稳定性。此外，在低规律性要求下，它们的时间误差估计也成功得出，即半分化问题的精确解决方案仅被认为是连续的。数值结果表明，所提出的LRI方案比经典的指数时间差异方案更准确，并且具有更好的收敛速率，尤其是当界面参数接近零时。
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.
