rkdg and Hybrid rkdg+hweno Schemes for Divergence-Free   Magnetohydrodynamics Part I
  Balsara (2001) showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD was shown in Balsara (2004). In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free RKDG and hybrid RKDG+HWENO schemes for MHD that have accuracies that are better than second order. Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent one-dimensional tests are also presented showing that the schemes perform well on those tests. 
D. S. Balsara