Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Ideal magnetohydrodynamic (MHD) equations consist of a set of nonlinear hyperbolic conservation laws, with a divergence-free constraint on the magnetic field. Neglecting this constraint in the design of computational methods may lead to numerical instability or nonphysical features in solutions. In our recent work [F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics 230 (2011) 4828–4847], second and third order exactly divergence-free central discontinuous Galerkin methods were proposed for ideal MHD equations. In this paper, we further develop such methods with higher order accuracy. The novelty here is that the well-established H(div)-conforming finite element spaces are used in the constrained transport type framework, and the magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence-free magnetic field. The overall algorithm is local, and it can be of arbitrary order of accuracy. Numerical examples are presented to demonstrate the performance of the proposed methods especially when they are fourth order accurate.