变系数MHD-Boussinesq方程的二阶BDF时间离散格式的误差分析
Error Analysis of a Second-Order BDF Time Discretization Scheme for the Variable-Coefficient MHD-Boussinesq Equations
摘要: 基于外推线性化方法，本文研究了具有变系数的二维不可压缩磁流体动力学方程组和热对流扩散方程所耦合MHD-Boussinesq方程组的二阶BDF时间离散格式，理论上证明了该格式具有无条件稳定性。在合理的正则性假设下，给出了速度、压力、磁场和温度在离散范数意义下的最优时间收敛阶，证明了格式具有二阶时间精度。最后通过数值实验验证了理论分析的结果。
Abstract: Based on the extrapolated linearization method, this paper investigates a second-order BDF time-discrete scheme for the coupled MHD-Boussinesq system, which consists of the two-dimensional incompressible magnetohydrodynamic equations with variable coefficients and the heat convection-diffusion equation. The unconditional stability of the scheme is theoretically proven. Under reasonable regularity assumptions, optimal temporal convergence orders in the discrete norm are derived for the velocity, pressure, magnetic field, and temperature, demonstrating that the scheme achieves second-order temporal accuracy. Finally, numerical experiments are conducted to validate the theoretical findings.
文章引用：徐轩. 变系数MHD-Boussinesq方程的二阶BDF时间离散格式的误差分析[J]. 应用数学进展, 2026, 15(2): 65-79. https://doi.org/10.12677/aam.2026.152050

参考文献

[1] Barleon, L., Casal, V. and Lenhart, L. (1991) MHD Flow in Liquid-Metal-Cooled Blankets. Fusion Engineering and Design, 14, 401-412. [Google Scholar] [CrossRef] 
[2] Hashizume, H. (2006) Numerical and Experimental Research to Solve MHD Problem in Liquid Blanket System. Fusion Engineering and Design, 81, 1431-1438. [Google Scholar] [CrossRef] 
[3] Smolentsev, S., Moreau, R., Bühler, L. and Mistrangelo, C. (2010) MHD Thermofluid Issues of Liquid-Metal Blankets: Phenomena and Advances. Fusion Engineering and Design, 85, 1196-1205. [Google Scholar] [CrossRef] 
[4] Davidson, P.A. (2001). An Introduction to Magnetohydrodynamics. Cambridge University Press.[CrossRef] 
[5] Titterton, D. and Weston, J.L. (2004) Strapdown Inertial Navigation Technology. Aerospace & Electronic Systems Magazine IEEE, 20, 33-34.
[6] Majda, A.J. and Bertozzi, A.L. (2001) Vorticity and Incompressible Flow. Cambridge University Press. [Google Scholar] [CrossRef] 
[7] Gao, H.D. and Qiu, W.F. (2019) A Linearized Energy Preserving Finite Element Method for the Dynamical Incompressible Magnetohydrodynamics Equations. Computer Methods in Applied Mechanics and Engineering, 346, 982-1001.
[8] Li, Y. and Luo, X. (2019) Second-Order Semi-Implicit Crank-Nicolson Scheme for a Coupled Magnetohydrodynamics System. Applied Numerical Mathematics, 145, 48-68. [Google Scholar] [CrossRef] 
[9] Zhang, Y.H., Hou, Y.R. and Shan, L. (2015) Numerical Analysis of the Crank-Nicolson Extrapolation Time Discrete Scheme for Magnetohydrodynamics Flows. Numerical Methods for Partial Differential Equations, 31, 2169-2208. [Google Scholar] [CrossRef] 
[10] Wang, S.H. and Li, Y. (2024) Optimal Convergent Analysis of a Linearized Euler Finite Element Scheme for the 2D Incompressible Temperature-Dependent MHD-Boussinesq Equations. Communications in Nonlinear Science and Numerical Simulation, 138, Article 108264. [Google Scholar] [CrossRef] 
[11] Heywood, J.G. and Rannacher, R. (1990) Finite-Element Approximation of the Nonstationary Navier-Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization. SIAM Journal on Numerical Analysis, 27, 353-384. [Google Scholar] [CrossRef] 
[12] Zhang, G.D., Yang, J. and Bi, C. (2018) Second Order Unconditionally Convergent and Energy Stable Linearized Scheme for MHD Equations. Advances in Computational Mathematics, 44, 505-540. [Google Scholar] [CrossRef] 
[13] Bian, D. and Liu, J. (2017) Initial-Boundary Value Problem to 2D Boussinesq Equations for MHD Convection with Stratification Effects. Journal of Differential Equations, 263, 8074-8101. [Google Scholar] [CrossRef] 

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