Abstract A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for solving two-dimensional hydrodynamic problems in cell-centered updated Lagrangian formulation. This method is the Lagrangian limit of the unsplit rDG-ALE formulation, and is obtained by assuming the equality of the grid velocity to the fluid velocity only at cell boundaries. The conservative variables and the Taylor basis defined on the time-dependent moving mesh, provide the piece-wise polynomial expansion in the updated Lagrangian formulation. A multi-directional nodal Riemann solver is implemented for computing the grid velocity at the vertices and the numerical flux at the cell boundaries. A characteristic limiting procedure is extended from the primitive variable version to the conservative variable version, and its performance is compared with the limiter on physical variables. A number of benchmark test cases are conducted to assess the accuracy, robustness, and non-oscillatory property of the DG(P0), DG(P1) and rDG(P1P2) methods. The numerical experiments demonstrate that the developed rDG method is able to attain the designed order of accuracy and the characteristic limiting procedure outperforms the limiter on physical variables in terms of the monotonicity and symmetry preservation for shock problems.

中文翻译：

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拉格朗日公式中可压缩流动的重构不连续伽辽金方法

摘要 为了解决以细胞为中心的更新拉格朗日公式中的二维流体动力学问题，开发了一种高阶精确重构不连续伽辽金 (rDG) 方法。该方法是未分裂 rDG-ALE 公式的拉格朗日极限，通过假设网格速度与仅在单元边界处的流体速度相等而获得。在瞬态移动网格上定义的保守变量和泰勒基础，在更新的拉格朗日公式中提供分段多项式展开。实现了多方向节点黎曼求解器，用于计算顶点处的网格速度和单元边界处的数值通量。一个特征限制程序从原始变量版本扩展到保守变量版本，并将其性能与物理变量限制器进行比较。进行了许多基准测试案例以评估 DG(P0)、DG(P1) 和 rDG(P1P2) 方法的准确性、稳健性和非振荡特性。数值实验表明，所开发的 rDG 方法能够达到设计的精度顺序，并且特征限制程序在冲击问题的单调性和对称性保持方面优于物理变量限制器。