用于三维变密度不可压无感应磁流体力学方程的高阶能量稳定且电荷守恒的拉格朗日有限元方法

中文摘要 AI 辅助

本文研究带有移动界面的变密度不可压无感应磁流体力学（MHD）方程的求解问题。我们在拉格朗日框架下提出了一种高阶、能量稳定且电荷守恒的混合有限元方法。该方法利用移动高阶曲面网格技术来捕捉界面，对速度场的时间离散采用二阶向后差分公式（BDF2），对网格控制点的更新采用二阶亚当斯-巴什福思方法（AB2）。速度-压力对采用带grad-div稳定化的高阶等参泰勒-胡德单元。此外，为确保离散电荷守恒，采用高阶参数化H(div)-符合BDM单元来处理电流密度。我们证明了全离散格式的无条件离散能量稳定性。数值实验验证了该算法的高阶精度和无条件能量稳定性。

英文摘要

In this paper, we develop a high-order, energy-stable and charge-conservative Lagrangian finite element method for variable-density incompressible inductionless magnetohydrodynamic (MHD) equations. The method utilizes the moving high-order curved tetrahedral mesh to track the material interface. Second-order Backward Differentiation Formula (BDF2) is used for the temporal discretization of material derivative, together with the second-order Adams--Bashforth method (AB2) for the update of control points of the meshes. High-order isoparametric Taylor-Hood elements with grad-div stabilization are used for the velocity-pressure pair. While, to ensure the discrete charge conservation, high-order parametric $\BH(\Div)$-conforming element is adopted for the current density. In the absence of external force, the unconditional energy-stability of the fully discrete scheme is proven. Finally, 3D numerical experiments are conducted to confirm the expected high-order accuracy for smooth solutions, the energy stability property and the capability of the proposed method.

发表机构

• School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）
• Academy of Mathematics and Systems Science, Xinjiang University（新疆大学数学与系统科学研究院）
• State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University（西安交通大学航天学院）

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