不可压磁流体力学方程组的高精度数值解法

High Precision Numerical Method for the Incompressible Magnetohydrodynamics Equations

为数值求解低雷诺数下不可压流体在电磁场作用下的流动,提出一种四阶紧致差分方法.由二维原始变量的MHD方程组出发,推导出具有较少未知量的电流密度-涡量-流函数形式MHD方程组.建立了求解二维非定常不可压MHD方程组的电流密度-涡量-流函数形式的四阶精度紧致差分格式.为验证本文提出的高精度紧致差分方法的精确性和可靠性,对有解析解的二维非定常不可压MHD方程组的初边值问题进行数值模拟,数值结果证明本文所建立的高阶紧致格式精确有效并且无条件稳定.

In this study, a four-order compact difference method for incompressible fluid flows in the presence of a magnetic field at low magnetic Reynolds number is proposed. Through the two-dimensional primitive variable MHD equations, we get the current density-vorticity-stream function MHD equations which have less variables. Then the fourth-order compact difference schemes is developed for solving two-dimensional time-dependent incompressible current density-vorticity-stream function MHD equations. In order to prove the accuracy and reliability of the high-order compact difference method, a numerical experiment with exact solutions to the two-dimensional time-dependent incompressible MHD equations with initial and boundary conditions is given. The numerical results demonstrate that the present high-order compact method is accurate, effective and unconditionally stable.