This paper investigates sparse grids on a hexagonal cell structure using a Local-Galerkin method (LGM) or generalized spectral element method (SEM). Such methods allow sparse grids to be used, known as serendipity grids in square cells. This means that not all points of the full grid are used. Using a high-order polynomial, some points of each cell are eliminated in the discretization, and thus saving Central Processing Unit (CPU) time. Here a sparse SEM scheme is proposed for hexagonal cells. It uses a representation of fields by second-order polynomials and achieves third-order accuracy. As SEM, LGM is strictly local for explicit time integration. This makes LGM more suitable for multiprocessing computers compared with classical Galerkin methods. The computer time depends on the possible timestep and program implementation. Assuming that these do not change when going to a sparse grid, the potential saving of computer time due to sparseness is 1:2. The projected CPU saving in 3-D from sparseness is by a factor of 3:8. A new spectral procedure is used in this paper, called the implied spectral equation (ISE). This procedure allows for some collocation points to use any finite difference scheme of high order and the time derivatives of other spectral coefficients are implied.

中文翻译：

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平面内匀速平流的六边形单元上的三阶稀疏网格广义谱元

本文使用局部伽辽金法 (LGM) 或广义光谱元素法 (SEM) 研究六角形单元结构上的稀疏网格。此类方法允许使用稀疏网格，称为方形单元格中的意外网格。这意味着并非整个网格的所有点都被使用。使用高阶多项式，在离散化中消除了每个单元格的某些点，从而节省了中央处理单元 (CPU) 时间。这里提出了一种用于六边形单元的稀疏 SEM 方案。它使用二阶多项式表示场并达到三阶精度。作为 SEM，LGM 对于显式时间积分是严格本地的。与经典的伽辽金方法相比，这使得 LGM 更适合多处理计算机。计算机时间取决于可能的时间步长和程序实现。假设这些在进入稀疏网格时不会改变，由于稀疏而潜在的计算机时间节省是 1:2。由于稀疏而在 3-D 中预计的 CPU 节省是 3:8。本文使用了一种新的光谱程序，称为隐含光谱方程 (ISE)。这个过程允许一些搭配点使用任何高阶的有限差分方案，并且暗示了其他谱系数的时间导数。