A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Magnetohydrodynamics Equations

We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$\alpha$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We numerically examine the effectiveness of our method using a manufactured solution and observe our method yields optimal convergence rates in the $L_2$ norm for the velocity field, pressure field, magnetic field, and magnetic pressure field. We further find our method is pressure robust. We then apply our method to a selection of benchmark problems and numerically confirm our method is energy stable.