In this paper, a high-order direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is developed for compressible fluid flows in two-dimensional (2D) Cartesian and cylindrical coordinates. The scheme in 2D cylindrical coordinates is based on the control volume approach and it can preserve the conservation property for all the conserved variables including mass, momentum and total energy. In this hydrodynamic scheme, a kind of high-order Taylor expansion basis function on the general element is used to construct the interpolation polynomials of the physical variables for the DG discretization. The terms including the material derivatives of the test functions are omitted, which simplifies the scheme significantly. Furthermore, the mesh velocity in the direct ALE framework is obtained by implementing an adaptive mesh movement method with a kind of dimensional-splitting type monitor function. This type of mesh movement method can automatically concentrate the mesh nodes near the regions with large gradients of the variables, which can greatly improve the resolutions of numerical solutions near the specified regions. For removing the numerical oscillations in the simulations, a Hermite Weighted Essential Non-oscillatory (HWENO) reconstruction is employed as a slope limiter. Finally, some test cases are displayed to verify the accuracy and the good performance of our scheme.

中文翻译：

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二维直角坐标和柱坐标下可压缩流的高阶任意拉格朗日-欧拉间断伽辽金方法


在本文中，针对二维（2D）笛卡尔和柱坐标系中的可压缩流体流动，开发了一种高阶直接任意拉格朗日-欧拉（ALE）不连续伽辽金（DG）格式。二维柱坐标系中的方案基于控制体积方法，它可以保留所有守恒变量（包括质量、动量和总能量）的守恒性质。在该流体动力学方案中，使用一种通用单元上的高阶泰勒展开基函数来构造物理变量的插值多项​​式以进行DG离散化。包括测试函数的材料导数在内的术语被省略，这显着简化了方案。此外，直接 ALE 框架中的网格速度是通过采用一种具有维数分割型监控功能的自适应网格移动方法来获得的。这种网格移动方法可以自动将网格节点集中在变量梯度较大的区域附近，从而可以大大提高指定区域附近数值解的分辨率。为了消除模拟中的数值振荡，采用 Hermite 加权基本非振荡 (HWENO) 重建作为斜率限制器。最后，展示了一些测试用例来验证我们方案的准确性和良好性能。