In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (--) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.

中文翻译：

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时变不可压缩 MHD 方程的完全解耦、线性化和稳定有限体积法


在本文中，我们考虑基于有限体积法的瞬态不可压缩磁流体动力学方程的完全解耦和线性化数值格式的稳定性和收敛性。采用最低等阶混合有限元对(--)来近似速度、压力和磁场，并引入压力投影稳定来绕过离散inf-sup条件的限制。采用半隐式处理对非线性项进行线性化，并采用一阶投影方案对速度和压力进行分解，形成一系列完全解耦和线性化的子问题。从理论分析的角度，给出了空间半离散格式和时空全离散格式数值解的一些新颖的稳定性结果，并通过使用能量法和选择不同的测试函数给出了最优误差估计。从计算结果来看，完全解耦线性化的数值方案不仅保持了良好的精度，而且节省了大量的计算成本。