A Reconstructed Discontiuous Galerkin Method for the Magnetohydrodynamics on Arbitrary Grids

A reconstructed discontinuous Galerkin (RDG) method based on a Hierarchical Weighted Essentially Non-oscillatory (WENO) reconstruction using a Taylor basis, designed not only to enhance the accuracy of discontinuous Galerkin methods but also to ensure the nonlinear stability of the RDG method, is developed for the solution of the magnetohydro dynamics (MHD) on arbitrary grids. In this method, a quadratic polynomial solution (P2) is first reconstructed using a Hermite WENO (HWENO) reconstruction from the underlying linear polynomial (P 1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only Von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The gradients (first moments) of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the RDG method. Temporal discretization is done using a 4th order explicit Runge-Kutta method. The HLLD Riemann solver, introduced in the literature for one dimensional MHD problems, is extended to three dimensional problems on unstructured grids and used to compute the flux functions at interfaces in the present work. Divergence free constraint is satisfied using the so-called Locally Divergence Free (LDF) approach. The LDF formulation is especially attractive in the context of DG methods, where the gradients of independent variables are handily available and only one of the computed gradients needs simply to be modified by the divergence-free constraint at the end of each time step. The developed RDG method is used to compute a variety of fluid dynamics and magnetohydrodynamics problems in one, two, and three dimensions on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments indicate that this RDG(P1P 2) is able to achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method, and outperforms the third-order DG method (DG (P2)) in terms of both computing costs and storage requirements.