论文标题
霍尔MHD方程的结构保存和旋转有限元近似和预处理
Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations
论文作者
论文摘要
我们为不可压缩的，电阻的HALL磁性水力学（MHD）方程开发了结构性的有限元方法。这些方程式包含了欧姆定律中的霍尔当前项，并提供了比标准MHD方程的完全离子等离子体的描述，其长度比离子皮肤深度接近或更小。我们引入了HALL MHD的固定离散分离公式，该公式构成了磁性高斯定律（符合求解器的公差），并证明了Picard线性化的适当性和收敛性。对于瞬态问题，我们提出时间离散化，以保留能量和磁性和混合螺旋性，正好在两种类型的边界条件下的理想极限。此外，我们为固定案例和瞬态案例提供了一种增强的拉格朗日预处理技术。我们通过几个数值实验证实了我们的发现。
We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.
