A superconvergent hybridizable discontinuous Galerkin method for weakly compressible magnetohydrodynamics

Abstract

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.