论文标题
最大结合原理保留半连接抛物线方程的集成因子runge-kutta方法
Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations
论文作者
论文摘要
大量的半连接抛物线方程满足最大结合原理（MBP）的意义，即时间依赖性解决方案在任何时候保留了由其初始和边界条件施加的均匀键的均匀束缚。近年来，对这些方程式的数值方程的调查引起了人们的关注，特别是对于时间离散。在本文中，我们通过集成因子runge-kutta（IFRK）方法研究高阶MBP保留时间集成方案。从半线性抛物线方程的空间散落系统开始，我们以一般形式介绍IFRK方法，并得出保留MBP的方法的足够条件。特别是，我们表明经典的四阶，四阶IFRK方案对某些典型的半线性系统进行了MBP保护，尽管并不保持强稳定性，可以立即将其应用于Allen-CAHN类型的方程式。此外，这些数值方案的错误估计值在理论上和数字上得到证明，以及通过长期进化行为的模拟来证明它们的效率。
A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Investigation on numerical schemes of these equations with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP-preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. In addition, error estimates for these numerical schemes are proved theoretically and verified numerically, as well as their efficiency by simulations of long-time evolutional behavior.
