二维Allen-Cahn方程的Crank-Nicolson型差分格式  

Crank-Nicolson Difference Scheme for Two Dimensional Allen-Cahn Equation

作　　者：张鑫 金元峰[1] 乔寒月 李春花[1] ZHANG XIN;JIN YUANFENG;QIAI HANYUE;LI CHUNHUA(Department of Mathematics,Yanbian University,Yanji 133002,China)

机构地区：[1]延边大学理学院数学系,延吉133002

出　　处：《应用数学学报》2021年第2期238-250,共13页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金项目(11761074);吉林省科技厅项目(20200301053RQ);吉林省自然科学基金项目(2020122336JC)资助。

摘　　要：本文主要研究相场模拟中的Allen-Cahn模型,考虑二维非线性Allen-Cahn方程,建立Crank-Nicolson差分格式,并给出截断误差.运用Browder定理,证明差分格式数值解的存在性.同时,利用截断函数法证明了差分格式在无穷范数下二阶无条件收敛.最后,对离散最大值原理进行研究说明,给出数值算例.In this paper,we study the Allen-Cahn model of the phase-field simulation.Considering the two-dimensional nonlinear Allen-C ahn equation,we establish Crank-Nicolson difference scheme,and give truncation errors.The existence of the difference solution is proved with the help of Browder fixed point theorem.At the same time,the difference scheme is demonstrated to be unconditionally convergent in L∞ norm by introducing an auxiliary smooth function.In the end,the study of maximum principle is given.The numerical experiments also verify the reliability of the method.

关 键 词：Allen-Cahn方程 CRANK-NICOLSON格式 收敛性 最大值原理 数值算例 

分 类 号：O212.7[理学—概率论与数理统计]