Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2024-05-15 DOI:10.1016/j.apnum.2024.05.009

Guomei Zhao , Shuaifei Hu

引用次数: 0

Abstract

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate $O (τ^{2})$ in $H^{1}$ - norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

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方形相场晶体模型无条件能量稳定二阶 BDF 方案的时间误差分析

本文首先提出并研究了方形相场晶体模型六阶非线性抛物线问题的二阶时间离散数值方案。然后，我们演示了具有质量守恒和能量耗散的两步反向微分公式（BDF-2）方案，其中高阶非线性项被隐式处理。此外，我们还提出了严格的误差分析，并证明了 H1 规范下的最优二阶收敛率 O(τ2)，其中 τ 是时间步长。最后，我们提供了一些数值结果来证实我们的理论分析。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.