数学
消散
点式的
艾伦-卡恩方程
有界函数
法学
类型(生物学)
标量(数学)
最大值原理
上下界
边界(拓扑)
应用数学
能量(信号处理)
数学分析
最优控制
数学优化
物理
几何学
统计
热力学
生物
生态学
政治学
作者
Lili Ju,Xiao Li,Zhonghua Qiao
标识
DOI:10.1007/s10915-022-01921-9
摘要
It is well-known that the Allen–Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solutions for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen–Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen–Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.
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