不可压缩矢量势磁流体力学方程组的一阶投影方法的时间收敛性

Temporal Convergence of the First-Order Projection Method for the Incompressible Vector Potential Magnetohydrodynamic System

本文研究了求解三维不可压缩矢量势磁流体力学方程组的一阶投影时间离散算法。该方程组是将原磁流体力学方程组中的磁场B写成旋度形式,即引入B = curlA。通过构造矢量势磁流体力学方程组的数值算法,使得磁场的数值解在全离散层面满足无散度条件。本文主要通过构造一阶投影格式,使得速度场的数值解也满足无散度条件,且所构造的投影格式对于任意时间步长都是无条件稳定的。在合理的正则性假设下,我们得到了速度和磁矢量势的一阶时间收敛阶。最后,通过数值算例验证了收敛性结果。In this paper, we consider a first-order projection finite element scheme for the three-dimensional incompressible magnetohydrodynamic system. This system of equations is to write the magnetic field B in the original magnetohydrodynamic equations in the curl form, which introduces B = curlA. By constructing the numerical algorithm of the system, the numerical solution of the magnetic field satisfies the divergence-free condition in fully discrete level. In this paper, by constructing the first-order projection scheme so that the numerical solution of the velocity field satisfies the divergence-free condition, and the constructed projection scheme is unconditionally stable for any time step. Under a reasonable regularity assumption, we derive the first-order temporal convergence order of the velocity and magnetic vector potential. Finally, the convergence results are verified by numerical examples.