On Explicit Discontinuous Galerkin Methods For Conservation Laws

The concept of projection applied to explicit discontinuous Galerkin (DG) schemes is investigated. The two explicit DG methods in this study are based on a 'predictor-corrector' formulation, the first introduced by Lorcher, Gassner, and Munz called space-time expansion discontinuous Galerkin or STE-DG scheme (Lorcher et al. 2007, Gassner et al. 2008), and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction takes place; the two methods differ, however, in the evaluation of the projections in the space-time volume integrals. The STE-DG scheme evaluates these in a straightforward manner, whereas the moment scheme employs a successive procedure with each moment update uses the results by the lower-order updates. The trade-offis that the moment scheme has the disadvantage of a more elaborate corrector step and the significant advantage of a CFL (Courant-Friedrichs-Lewy) condition of 1 for all (degree) pand accuracy order of 2p + 1 (super accuracy property) for one-dimensional (1D) advection. In contrast, the STE-DG method is accurate to only the expected order of p + 1 and has a more restrictive CFL condition. Due to the predictor-corrector formulation that does not involve methods of characteristics, these schemes extend easily to systems of equations in multiple dimensions. Concerning the case of two spatial dimensions (2D), for an advection using a Cartesian grid, when the flow does not align with the axes, especially when it is along a diagonal direction, the CFL conditions for the moment schemes also become restrictive and need improvement as will be shown by Fourier (von Neumann) analyses. Published by Elsevier Ltd.