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.
翻译:大型半线性抛物线方程式符合最大约束原则(MBP),因为时间依赖的解决方案在任何时间保留其初始条件和边界条件所施加的统一点约束。关于这些方程式中保留MBP的数值办法的调查近年来日益引起注意,特别是对于时间分解而言。在本文件中,我们研究高调的MBP-保留时间的整合办法,方法是综合因子Runge-Kutta(IFRK)方法。从半线性抛物方程式的空间分解系统开始,我们以一般形式提出IFRK法方法,并为维护MBP法的方法提出充分的条件。特别是,我们表明典型的四阶段第四级IFRK法办法为某些典型的半线性半线性系统保留了MBP,尽管这种半线性办法可以立即适用于Allen-Cahn型方程式。此外,这些数值方法的误差估计数在理论上和数字上证明是经过核实的,并且通过对长期进化行为进行模拟的效率。