简化磁流体方程二阶向后差分伽辽金有限元方法

BDF2 Galerkin Finite Element Method for the Simplified Magnetohydrodynamic Flows

摘要: 本文针对低磁雷诺数下的简化磁流体（MHD）方程，提出了一种高效的线性化二阶时步伽辽金有 限元方法。 该方法时间离散采用二阶向后差分格式（BDF2）保证时间精度，针对方程中的非线 性对流项，设计了 Adams-Bashforth （AB）外推处理格式，将原非线性问题转化为线性求解 模式，实现了每个时间步仅需求解一个线性方程组，避免了非线性迭代，大幅降低了计算复杂度。 空间离散采用满足 inf-sup 条件的混合有限元方法，对速度、 压力、 电势进行协调逼近。 本文严 格证明了格式的无条件稳定性，推导得到了速度、压力、电势的最优二阶误差估计。

Abstract: An efficient linearized second-order time-stepping Galerkin finite element method is developed for simplified magnetohydrodynamic (MHD) flows in the low magnetic Reynolds number regime. The method uses the second-order backward difference formula (BDF2) for temporal discretization and an Adams-Bashforth (AB) extrapo- lation technique for the nonlinear convective terms, which linearizes the problem and yields only one linear system to solve at each time step, thus eliminating nonlinear it- erations and lowering computational cost substantially. A stable mixed finite element pair satisfying the inf-sup condition is used for the conforming spatial discretization of velocity, pressure and electric potential. The unconditional stability of the scheme is established, and optimal second-order error estimates for all three variables are obtained through rigorous mathematical analysis.