A Discontinuous Galerkin Method for the Magnetohydrodynamics on Arbitrary Grids

A discontinuous Galerkin method based on a Taylor basis is presented for the solution of the magnetohydrodynamics equations on arbitrary grids. Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell. Consequently, this formulation is able to provide a unified framework, where both cell-centered and vertex-centered finite volume schemes can be viewed as special cases of this discontinuous Galerkin method by choosing reconstruction schemes to compute the derivatives, offer the insight why the DG methods are a better approach than the finite volume methods based on either TVD/MUSCL reconstruction or essentially non-oscillatory (ENO)/weighted essentially non-oscillatory (WENO) reconstruction, and has a number of distinct, desirable, and attractive features, which can be effectively used to address some of shortcomings of the DG methods. The extension of the HLLD scheme to multi-dimensional problems is developed to compute fluxes across cell interfaces. The developed method is used to compute a variety of test cases. The numerical experiments demonstrate the high accuracy of this discontinuous Galerkin method.