Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes
Maximum Bound Principles for a Class of Semilinear Parabolic Equations and Exponential Time-Differencing Schemes
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一类半线性抛物型方程的最大界原理和指数时差格式
DOI：
10.1137/19m1243750
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发表时间：
2021-06-01
期刊：
影响因子：
10.2
通讯作者：
Qiao，Zhonghua
中科院分区：
文献类型：
--
作者：
Du，Qiang;Ju，Lili;Qiao，Zhonghua
半线性抛物方程的普遍性从其从物理学和生物学到材料和社会科学的众多应用中显而易见。本文考虑了一类抽象形式的半线性抛物方程的一个实用性质，即时不变最大界原理，即它的时变解在任何时候都保持由其初始条件和边界条件所赋予的一致逐点绝对值有界.首先，我们研究了一个分析框架的充分条件上，并导致这样一个最大限度的约束原则的时间连续的动态系统的无限或有限维。然后，我们利用一个合适的指数时间差分方法与适当选择的发电机的收缩半群开发一阶和二阶精确的时间离散计划，满足最大限度的原则，无条件地在时间离散设置。所提出的方案的误差估计沿着他们的能量稳定性。向量和矩阵值系统的扩展也进行了讨论。我们证明，这里开发的抽象框架和分析技术提供了一个有效的和统一的方法来研究抽象的演化方程，涵盖了各种各样的知名模型和它们的数值离散化方案的最大界原理。最后通过数值实验对理论结果进行了验证。
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, witha linear dissipative operator anda nonlinear operator in space, namely, a time-invariant maximum bound principle, in the sense that the time-dependent solutionpreserves 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 onandthat 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.