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.

► Locally divergence-free central DG methods are proposed and numerically investigated for ideal MHD equations.
► They locally enforce a divergence-free constraint on the magnetic field. This strategy is simple to implement.
► Divergence-free treatment is important for stable numerical MHD simulations.
► The proposed methods demonstrate good stability and designed high order accuracy.
► Comparison is made to advance the understanding on divergence-free treatments in DG and central DG frameworks.