AI 中文总结
研究针对Cahn-Hilliard-Navier-Stokes模型，采用标量辅助变量方法，开发出线性、完全解耦且无条件能量稳定的全离散有限元格式，经理论分析与数值实验验证了格式的稳定性、误差估计及有效性。
AI 中文摘要
本文通过采用标量辅助变量（SAV）方法，为Cahn-Hilliard-Navier-Stokes（CHNS）系统开发了一种线性、完全解耦且无条件能量稳定的全离散有限元格式。与现有基于SAV的公式不同，我们仅引入一个标量辅助变量及新的更新方式来重新表述CHNS方程中的非线性项。采用隐式-显式（IMEX）欧拉格式进行时间离散，有限元方法进行空间离散。所得全离散格式可有效分解为两个线性子问题和一个标量二次代数方程，简化了实现。证明了该格式满足无条件离散能量耗散律并建立了相关稳定性。还推导了全离散有限元逼近的最优阶\(L^2\)误差估计。最后通过数值实验验证了理论结果并展示了方法的有效性。
英文摘要
In this paper, we develop a linear, fully decoupled, and unconditionally energy-stable fully discrete finite element scheme for the Cahn--Hilliard--Navier--Stokes (CHNS) system by employing the scalar auxiliary variable (SAV) approach. Unlike existing SAV-based formulations that typically introduce multiple auxiliary variables or additional techniques to handle different nonlinearities, we introduce only one scalar auxiliary variable together with a novel update of the auxiliary variable to reformulate all nonlinear terms arising from the Cahn--Hilliard and Navier--Stokes equations, yielding an equivalent reformulation of the original CHNS system. An implicit--explicit (IMEX) Euler scheme is applied for temporal discretization, where the linear terms are treated implicitly and the nonlinear terms explicitly, while a finite element method is adopted for spatial discretization. The resulting fully discrete scheme can be efficiently decomposed into two linear subproblems and one scalar quadratic algebraic equation, which significantly simplifies the implementation. Furthermore, we prove that the proposed scheme satisfies an unconditional discrete energy dissipation law and establish its stability with respect to several relevant norms. Optimal-order $L^2$ error estimates are also derived for the fully discrete finite element approximation. Finally, a series of numerical experiments are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.
Comments27 pages, 5 figures
