晶体相场方程的线性化Crank-Nicolson格式的误差分析

doi:10.6040/j.issn.1671-9352.0.2018.146

摘要/Abstract

摘要： 晶体相场模型是一类空间六阶非线性发展方程。首先,给出了线性化Crank-Nicolson格式,该格式在第一、二时间层是显式差分格式,其余时间层是线性化隐式差分格式。在建立差分格式的过程中,将非线性项(u3)_xx改写成(3u2 u_x)_x,利用中心差商对其进行离散。其次,证明了差分格式解的先验估计式及无条件收敛性,收敛阶在时空方向均为二阶。最后通过数值算例,验证差分格式是有效的。

关键词: 晶体相场方程, 线性化Crank-Nicolson 格式, 收敛性, 非线性问题, 线性化

Abstract: The phase field crystal model is a high order nonlinear evolutionary equation with the sixth order derivative in space. A linearized Crank-Nicolson scheme is presented. The scheme is explicit at the first-and second-time level. We just only to solve an implicit linearized scheme at the rest of the time level. In the derivation of the scheme, the nonlinear term (u3)_xx is rewritten to be (3u2 u_x)_x, and then be discretized by central difference quotient. The priori estimate of the numerical solution and unconditional convergence is proved in L2 norm. The convergence order is two in time and space. Some numerical examples are presented to demonstrate the theoretical results.

Key words: phase field crystal equation, linearized Crank-Nicolson scheme, convergence, nonlinear problem, linearization

中图分类号:

O241.82

李娟. 晶体相场方程的线性化Crank-Nicolson格式的误差分析[J]. 《山东大学学报(理学版)》, 2019, 54(6): 118-126.

LI Juan. Error analysis of a linearized Crank-Nicolson scheme for the phase field crystal equation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 118-126.

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