3维不可压缩MHD方程组的全离散隐式/显式差分有限元算法

Magnetohydrodynamic（MHD）is of important applications in the astrophysics,.controlled nuclear reaction and MHD dynamotor. First, we design the fully discrete.implicit/explicit method for solving the nonstationary incompressiblethe MHD.equations on a 3D bouned convex domain, time discretization is based on the.implicit/explicit difference scheme with the second accuracy, namely, the second.order stability implicit scheme is used to deal withthe linear terms in MHD.equations and the simple second order explicit scheme is used to deal with the.nonlinear terms in MHD equations. Next, we design the fully discrete.implicit/explicit difference finite element method for solving the nonstationary.incompressible MHD equations，where we use the diference method in z-direction and.the finite element method in the (x,y)-direction for solving the 2D Stokes.equations and the Maxwell equations. For the fully discrete difference finite.element methods for solving the 3D MHD equations, we will analyze the optimal H^1.error estimates of the numerical solution by the induction method and the optimal.L^2-error estimates by using the negative norm technique without using the.standard duality argument. Numerical analysis shows our method is of the.advantages of taking larger time step, simple computation and needing a little.save amount. Finally, design the computational programme and numerical test the.physical MHD problems. This research will provide some important contributions to.the nonlinear sciences and MHD numerical computations.

磁流体力学（MHD）在天体物理、受控热核反应和磁流体发电中具有重要的应用。首先设计求解3维有界凸区域上的非定常不可压MHD方程的时间变量离散的二阶隐式/显式差分格式（其线性项用稳定的二阶隐式格式，非线性项用计算简单的二阶显式格式）。三维MHD方程经过隐式显式时间离散后，在每个时间层仍然需要求解线性三维的Stokes方程和Maxwell方程，对此，我们将运用差分有限元方法求解它，即径向方向用差分格式， 这样在水平方向形成的二维Stokes方程和Maxwell方程，又可以用并行计算。对于求解三维MHD方程全离散差分有限元算法，用数学归纳法推导H^1最优误差估计，用负范数技巧推导出速度和磁强度的L^2最优误差。数值分析表明该算法具有可以取大时间步长，计算量少，存贮量少的优点，最后编制程序，数值模拟实际应用问题。该项目的研究将会为非线性科学和磁流体数值模拟的研究做出重要的贡献。