Abstract In this paper, a third-order reconstructed discontinuous Galerkin (DG) method based on a weighted variational minimization principle, which is denoted as P 1 P 2 (WVr) method, is presented for solving the incompressible flows on unstructured grids. In this method, the first-order degrees of freedom (DoFs) are obtained directly from the underlying second-order DG method, while the second-order DoFs are reconstructed through the weighted variational reconstruction. Specifically, we first introduce a weighted interfacial jump integration (WIJI) function which represents a measure of the jump between the reconstructed polynomial solutions from two neighboring cells. Then, we build the constitutive relations by minimizing this WIJI function using the variational method. A number of incompressible flow problems in both steady and unsteady forms are presented to assess the performance of the proposed P 1 P 2 (WVr) method. The numerical results demonstrate that the P 1 P 2 (WVr) method is able to achieve the designed optimal third-order accuracy at a significantly reduced computational costs. Moreover, when a suitable value of the weight parameter is chosen to be used, the P 1 P 2 (WVr) method outperforms the reconstructed DG methods based on either least-squares or Green-Gauss reconstruction for the simulations of incompressible flows.

中文翻译：

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求解不可压缩流的三阶加权变分重构不连续伽辽金方法

摘要 本文提出了一种基于加权变分最小化原理的三阶重构不连续伽辽金(DG)方法，称为P 1 P 2 (WVr)方法，用于求解非结构化网格上的不可压缩流动。在该方法中，一阶自由度 (DoF) 直接从底层的二阶 DG 方法中获得，而二阶自由度是通过加权变分重构来重构的。具体来说，我们首先引入加权界面跳跃积分 (WIJI) 函数，该函数表示来自两个相邻单元的重构多项式解之间跳跃的度量。然后，我们通过使用变分方法最小化这个 WIJI 函数来构建本构关系。提出了许多稳态和非稳态形式的不可压缩流动问题，以评估所提出的 P 1 P 2 (WVr) 方法的性能。数值结果表明，P 1 P 2 (WVr) 方法能够以显着降低的计算成本实现设计的最佳三阶精度。Moreover, when a suitable value of the weight parameter is chosen to be used, the P 1 P 2 (WVr) method outperforms the reconstructed DG methods based on either least-squares or Green-Gauss reconstruction for the simulations of incompressible flows.