This work proposes a hybridizable discontinuous Galerkin (HDG) method for the solution of magnetohydrodynamic (MHD) problems with weakly compressible flows. A novel fluid formulation that adopts the velocity and the pressure as primal variables is first derived and its superior properties, compared to alternative density–momentum-based approaches, are demonstrated on a simple benchmark. The coupled MHD formulation exhibits superconvergence properties for both the fluid velocity and the magnetic induction, a feature not present in any HDG formulation published in this field. An alternative MHD formulation, adopting a fluid-type solver for the solution of the magnetic subproblem, is also considered and its advantages and disadvantages are discussed. The convergence properties of the proposed formulations for the single physics and for the coupled problem are examined on an extensive set of numerical examples in both two and three dimensions, on structured and unstructured meshes and at low and high Hartmann numbers.

这项工作提出了一种可混合的不连续伽辽金 (HDG) 方法，用于解决具有弱可压缩流动的磁流体动力学 (MHD) 问题。一种采用速度和压力作为主要变量的新型流体公式首先被推导出来，与替代的基于密度-动量的方法相比，它的优越特性在一个简单的基准上得到了证明。耦合 MHD 公式对流体速度和磁感应都表现出超收敛特性，这一特征在该领域发表的任何 HDG 公式中都不存在。还考虑了另一种 MHD 公式，采用流体型求解器来解决磁性子问题，并讨论了其优缺点。