SIAM Review, Volume 63, Issue 2, Page 317-359, January 2021.
The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t={\mathcal{L}} u+f[u]$, with ${\mathcal{L}}$ a linear dissipative operator and $f$ a nonlinear operator in space, namely, a time-invariant maximum bound principle, in the sense that the time-dependent solution $u$ preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for sufficient conditions on ${\mathcal{L}}$ and $f$ that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then we utilize a suitable exponential time-differencing approach with a properly chosen generator of the contraction semigroup to develop first- and second-order accurate temporal discretization schemes that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to studying the maximum bound principle of the abstract evolution equation that covers a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results.

中文翻译：

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一类半线性抛物线方程和指数时差方案的最大有界原理

SIAM 评论，第 63 卷，第 2 期，第 317-359 页，2021 年 1 月。
从物理学和生物学到材料和社会科学的众多应用中，可以清楚地看出半线性抛物线方程的普遍性。在本文中，我们考虑了抽象形式 $u_t={\mathcal{L}} u+f[u]$ 的一类半线性抛物线方程的实际理想性质，其中 ${\mathcal{L}}$ a线性耗散算子和 $f$ 空间中的非线性算子，即时不变最大边界原则，在这个意义上，时间相关解 $u$ 始终保持由其初始值强加的绝对值的均匀逐点边界和边界条件。我们首先研究了 ${\mathcal{L}}$ 和 $f$ 上的充分条件的分析框架，这导致了无限或有限维的时间连续动态系统的最大界限原理。Then we utilize a suitable exponential time-differencing approach with a properly chosen generator of the contraction semigroup to develop first- and second-order accurate temporal discretization schemes that satisfy the maximum bound principle unconditionally in the time-discrete setting. 所提出方案的误差估计与其能量稳定性一起推导出来。还讨论了向量和矩阵值系统的扩展。我们证明，这里开发的抽象框架和分析技术提供了一种有效且统一的方法来研究涵盖各种知名模型及其数值离散化方案的抽象演化方程的最大界原理。还进行了一些数值实验来验证理论结果。