Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations

In this paper, we propose and numerically investigate a family of locally divergence-free central discontinuous Galerkin methods for ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods (SIAM Journal on Numerical Analysis 45 (2007) 2442-2467) for hyperbolic equations, with the use of approximating functions that are exactly divergence-free inside each mesh element for the magnetic field. This simple strategy is to locally enforce a divergence-free constraint on the magnetic field, and it is known that numerically imposing this constraint is necessary for numerical stability of MHD simulations. Besides the designed accuracy, numerical experiments also demonstrate improved stability of the proposed methods over the base central discontinuous Galerkin methods without any divergence treatment. This work is part of our long-term effort to devise and to understand the divergence-free strategies in MHD simulations within discontinuous Galerkin and central discontinuous Galerkin frameworks.