High-order finite element arbitrary Lagrangian-Eulerian resistive magnetohydrodynamics coupled to a multi-physics code

We present a high-order finite element, arbitrary Lagrangian-Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured 2D and 3D high-order meshes. For the case of 2D, we consider two polarizations of the magnetic field: in-plane and out-of-plane. In each case, the resistive magnetic induction equation is discretized using compatible mixed finite element methods which preserve the divergence-free nature of the magnetic field. We augment the magnetic diffusion equations with an additional scalar potential solve driven by voltage source boundary conditions to facilitate the coupling of the MHD model with an external circuit solver. Electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. The coupled magnetic diffusion and Lagrangian hydrodynamics equations are solved in time using a novel multirate implicit/explicit (IMEX) approach where the magnetic diffusion solve is performed implicitly over a large timestep while the Lagrangian hydrodynamics is solved with an explicit, energy conserving two-stage RK method. When the Lagrangian motion of the mesh causes significant distortion, that distortion is corrected with an optimization of the mesh, followed by an ALE remap step of the MHD state variables. To remap magnetic flux, we use a high-order constrained transport (CT) method to maintain the divergence-free nature of the magnetic field. Finally, we verify each stage of the discretization via a set of numerical experiments. Such an algorithm has applications to magnetically confined fusion and high-energy density physics.