2025年04月05日 星期六

广西师范大学学报（自然科学版） ›› 2025, Vol. 43 ›› Issue (2): 193-206.doi: 10.16088/j.issn.1001-6600.2024031701

• 数学与统计学 • 上一篇    下一篇

变密度MHD方程完全解耦且无条件能量稳定的数值算法

王兆伟, 王旦霞*   

  1. 太原理工大学 数学学院, 山西 晋中 030600
  • 收稿日期:2024-03-17 出版日期:2025-03-05 发布日期:2025-04-02
  • 通讯作者: 王旦霞(1979—), 女, 山西太原人, 太原理工大学教授, 博士。E-mail: 2621259544@qq.com
  • 基金资助:
    山西省基础研究计划(202203021211129); 山西省回国留学人员科研资助项目(2021-029);山西省国际合作平台项目(202104041101019)

Fully Decoupled and Unconditionally Energy-stable Schemes for MHD Equations with Variable Density

WANG Zhaowei, WANG Danxia*   

  1. School of Mathematics, Taiyuan University of Technology, Jinzhong Shanxi 030600, China
  • Received:2024-03-17 Online:2025-03-05 Published:2025-04-02

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摘要： 本文致力于对变密度不可压缩磁流体动力学方程建立全解耦且无条件能量稳定的数值算法。首先,引入辅助中间速度变量,分别与Gauge-Uzawa 方法的对流形式和守恒形式相结合,建立2个一阶半离散数值算法。2个算法成功地解耦所有耦合项,使得只需在离散级求解线性化子问题, 所以大大提高计算效率。其次,证明2个算法都是无条件能量稳定的,并且对流形式的有限元全离散化算法也是无条件能量稳定的。最后,用数值实验验证解耦方案的准确性和有效性。

关键词: 磁流体动力学, 全解耦, 无条件能量稳定, 变密度, Gauge-Uzawa方法

Abstract: This article is dedicated to establishing fully decoupled and unconditionally energy-stable numerical algorithms for the incompressible magnetohydrodynamics (MHD) equations with variable density. The overall idea is as follows: Firstly, two first-order semi-discrete numerical algorithms are developed based on the Gauge-Uzawa method in both convective and conserved forms. Since both algorithms successfully decouple all coupled terms, only the linearized subproblems need to be solved at the discrete level, which significantly improve computational efficiency. Secondly, it is demonstrated that both algorithms are unconditionally energy-stable, and the finite element fully discrete algorithm in convective form is also unconditionally energy-stable. Finally, numerical experiments confirm the accuracy and effectiveness of the decoupled schemes.

Key words: magnetohydrodynamics, fully decoupled, unconditionally energy-stable, variable density, Gauge-Uzawa method

中图分类号:  O242

[1] LI Y, AN R. Temporal error analysis of Euler semi-implicit scheme for the magnetohydrodynamics equations with variable density[J]. Applied Numerical Mathematics, 2021, 166: 146-167. DOI: 10.1016/j.apnum.2021.04.006.
[2] CHEN Hang, HE Y Y, CHEN H T. Stability and temporal error estimate of scalar auxiliary variable schemes for the magnetohydrodynamics equations with variable density[J]. Numerical Methods for Partial Differential Equations, 2024, 40(1): e23067. DOI: 10.1002/num.23067.
[3] 杨莉.求解Maxwell方程的间断有限元方法后处理技术研究[D].成都:电子科技大学,2022. DOI: 10.27005/d.cnki.gdzkn.2022.001713.
[4] HE Y N. Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations[J]. IMA Journal of Numerical Analysis, 2015, 35(2): 767-801. DOI: 10.1093/imanum/dru015.
[5] 胡凯伦, 陈敏, 罗宏. 磁流体方程全局吸引子的正则性[J].广西师范大学学报(自然科学版),2024,42(1):120-127. DOI: 10.16088/j.issn.1001-6600.2023032403.
[6] 薛媛媛, 江珊. 带Navier边界条件的广义随机Navier-Stokes方程解的适定性[J].吉首大学学报(自然科学版), 2024,45(1):13-18. DOI: 10.13438/j.cnki.jdzk.2024.01.003.
[7] 罗宏, 蒲志林, 周亚非.Navier-Stokes方程的渐近吸引子[J].广西师范大学学报(自然科学版),2009,27(2):38-41. DOI: 10.3969/j.issn.1001-6600.2009.02.010.
[8] LI C Y, LI Y. Optimal L2 error analysis of first-order Euler linearized finite element scheme for the 2D magnetohydrodynamics system with variable density[J]. Computers and Mathematics with Applications, 2022, 128: 96-107. DOI: 10.1016/j.camwa.2022.10.013.
[9] 毛静静. 变密度不可压缩Navier-Stokes方程的Gauge-Uzawa方法的误差估计[D].厦门:厦门大学,2019.
[10] 申肖娟. 非定常不可压磁流体方程压力投影有限元方法的研究[D].焦作:河南理工大学,2020.
[11] 刘盟. 向列相液晶流问题的能量稳定数值方法研究[D].太原:太原理工大学,2022.
[12] PYO J H, SHEN J. Gauge-Uzawa methods for incompressible flows with variable density[J]. Journal of Computational Physics, 2007, 221(1):181-197. DOI: 10.1016/j.jcp.2006.06.013.
[13] NOCHETTO R H, PYO J H. The Gauge-Uzawa finite element method. Part I: the Navier-Stokes equations[J]. SIAM Journal on Numerical Analysis, 2005, 43(3):1043-1068. DOI: 10.1137/040609756.
[14] NOCHETTO R H, PYO J H. The Gauge-Uzawa finite element method. Part II: the Boussinesq equations[J]. Mathematical Models and Methods in Applied Sciences, 2006, 16(10): 1599-1626. DOI: 10.1142/S0218202506001649.
[15] 张青. 非定常Magnetohydrodynamic方程的Gauge有限元方法[D].乌鲁木齐:新疆大学,2016.
[16] GERBEAU J F, LE BRIS C, LELIE｀VRE T. Mathematical methods for the magnetohydrodynamics of liquid metals[J]. Oxford: Oxford University Press, 2006. DOI: 10.1093/acprof:oso/9780198566656.001.0001.
[17] HECHT F. New development in FreeFem++[J]. Journal of Numerical Mathematics, 2012, 20(3/4): 251-266. DOI: 10.1515/jnum-2012-0013.
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