Cahn-Hilliard方程显式不变能量二次化方法(EIEQ)的精度和一致性

Accuracy and Consistency of the Explicit-Invariant Energy Quadratization Approach for the Cahn-Hilliard Equation

中文关键词:  Cahn-Hilliard方程  EIEQ  无条件能量稳定  完全解耦  松弛技巧

英文关键词:Cahn-Hilliard equation  EIEQ  unconditionally energy stable  fully decoupled  Relaxation technique

中文摘要:

      众所周知, EIEQ方法可以生成完全解耦、线性和无条件能量稳定的数值格式,因此受到许多研究人员的青睐.然而,不可否认的事实是,通过EIEQ方法获得的数值方法遵循“修改”的能量定律,而不是原始的能量.这主要是由于EIEQ方法中引入了一些辅助变量,截断误差会使辅助变量在数值计算过程中偏离原始定义.本文主要研究Cahn-Hilliard方程EIEQ方法的精度和一致性来解决这一问题.我们引入了一种辅助变量的松弛技术,并基于EIEQ构建了两种数值格式.分析结果表明新构建的数值格式不仅具有无条件能量稳定、线性和完全解耦的特点,而且可以有效地校正辅助变量和方程的原始能量定律.最后,给出几个二维和三维数值例子说明了新构建的数值格式的准确性和计算效率.

英文摘要:

      It is well known that the explicit-invariant energy quadratization (EIEQ) approach can generate fully decoupled, linear and unconditionally energy-stable numerical schemes, so it is favored by many researchers. However, the undeniable fact is that the numerical method obtained by EIEQ approach preserves the ``modified" energy law instead of the original energy. This is mainly due to the introduction of some auxiliary variables in EIEQ scheme, and the truncation error will make the auxiliary variables deviate from the original definition in the process of numerical calculation. The primary objective of this paper is to address this gap by providing the accuracy and consistency of the EIEQ method in the context of the Cahn-Hilliard equation. We introduce a relaxation technique for auxiliary variables and construct two numerical schemes based on EIEQ. The analysis results show that the newly constructed schemes are not only unconditionally energy stable, linear and fully decoupled, but also can effectively correct the errors introduced by auxiliary variables and follow the original energy law. Finally, several 2D and 3D numerical examples illustrate the accuracy and efficiency of the newly constructed numerical schemes.