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 Lörcher, Gassner, and Munz called space–time expansion discontinuous Galerkin or STE-DG scheme (Lörcher 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-off is 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) $p$ and 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.

研究了应用于显式不连续Galerkin（DG）方案的投影概念。这项研究中的两种明确的DG方法基于“预测校正器”公式，由Lörcher，Gassner和Munz首次提出，称为时空扩展不连续Galerkin或STE-DG方案（Lörcher等人，2007； Gassner等人作者（Huynh 2006，2013）等人独立介绍的第二种方法称为逆风矩方案。使用Cauchy-Kovalevsky过程，这两种方法的预测步骤基本相同，该过程不涉及相邻单元之间的数据交互。校正器步骤也共享相同的时空积分公式，并且是进行交互的地方。但这两种方法在时空体积积分的投影评估中有所不同。STE-DG方案以简单的方式评估这些值，而矩型方案采用连续过程，每个矩型更新使用低阶更新的结果。权衡的是，矩量法的缺点是校正步骤更加复杂，并且所有（度）CFL（Courant-Friedrichs-Lewy）条件为1的显着优势$p$ 和精度顺序 $\mathrm{2个}p+\mathrm{1个}$（超精度属性）用于一维（1D）对流。相比之下，STE-DG方法仅精确到预期的$p+\mathrm{1个}$并且具有更严格的CFL条件。由于预测器-校正器公式不涉及特征方法，因此这些方案可以轻松扩展到多维方程组。关于两个空间维度（2D），对于使用笛卡尔网格的对流，当流与轴不对齐时，尤其是沿对角线方向时，矩量方案的CFL条件也变得受限且需要傅立叶（von Neumann）分析将显示这种改进。