A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume/finite element scheme for the incompressible MHD equations
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.