We present a new divergence-free and well-balanced hybrid FV/FE scheme for the incompressible viscous and resistive MHD equations on unstructured mixed-element meshes in 2 and 3 space dimensions. The equations are split into subsystems. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms are solved at the aid of an explicit FV scheme, while the magnetic field is evolved in a divergence-free manner via an explicit FV method based on a discrete form of the Stokes law in the edges/faces of each primary element. To achieve higher order of accuracy, a pw-linear polynomial is reconstructed for the magnetic field, which is guaranteed to be divergence-free via a constrained L2 projection. The pressure subsystem is solved implicitly at the aid of a classical continuous FE method in the vertices of the primary mesh. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. This paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field lines.

翻译：我们提出了一种新的无散且精确平衡的混合FV/FE格式，用于求解二维和三维空间中非结构化混合单元网格上的不可压缩粘性电阻MHD方程组。该方程组被分解为若干子系统。压力定义在原始网格的顶点上，而速度场和磁场法向分量分别定义在二维和三维空间中基于边/面的对偶网格上。这允许以较为自然的方式满足速度场和磁场的无散条件。非线性对流项和粘性项通过显式FV格式求解，而磁场则通过基于每个原始单元边/面上斯托克斯定律离散形式的显式FV方法以无散方式演化。为实现高阶精度，采用约束L2投影重建满足无散性的磁场分段线性多项式。压力子系统通过原始网格顶点上的经典连续有限元方法隐式求解。为精确维持控制偏微分方程组的非平凡稳态平衡解（假设这些解是已知的），新算法的每一步均显式考虑已知平衡解，从而使该方法达到精确平衡。本文针对不同磁场存在下的顶盖驱动磁流体空腔问题进行了非常详尽的研究。最后，我们在多个简化的三维托卡马克位形中（即使在非常粗糙的、通常无需与磁力线对齐的非结构化网格上）展示了索洛维约夫平衡解的长时间模拟结果。