Abstract This article introduces a new vertex-centered quasi-Lagrangian discontinuous Galerkin method for two-dimensional compressible flows that is third-order accurate both in space and time. The computational domain is divided into structured quadrilateral cells with straight edges. Nodal control volumes with curved edges are constructed surrounding the grid vertices. The Euler equations in arbitrary Lagrangian-Eulerian (ALE) form are discretized on these nodal control volumes using a discontinuous Galerkin method. The time marching is implemented by the third-order strong-stability-preserving Runge-Kutta method. A compact HWENO reconstruction algorithm is used as limiter to eliminate spurious oscillations near discontinuities. The polynomial expression of fluid velocity defined in a nodal control volume is also obtained from the reconstruction procedure, and is used to calculate the moving velocity of corresponding grid vertex. In this way, the grid vertices are moved in a rigorously Lagrangian manner, although there are still mass fluxes between neighbouring nodal control volumes. Therefore the scheme is called as a quasi-Lagrangian one. The scheme is conservative for mass, momentum and total energy. Some numerical tests are carried out to demonstrate the accuracy and robustness of the scheme. It has a favorable qualitative behavior for discontinuous problems and optimal convergence rates for smooth problems.

中文翻译：

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二维可压缩欧拉方程的一种高阶顶点中心拟拉格朗日不连续伽辽金方法

摘要 本文介绍了一种新的二维可压缩流的以顶点为中心的拟拉格朗日不连续伽辽金方法，它在空间和时间上都具有三阶精度。计算域被划分为具有直边的结构化四边形单元。具有弯曲边缘的节点控制体积围绕网格顶点构建。任意拉格朗日-欧拉 (ALE) 形式的欧拉方程使用不连续伽辽金方法在这些节点控制体积上离散化。时间推进采用三阶强保稳 Runge-Kutta 方法实现。紧凑的 HWENO 重建算法用作限制器，以消除不连续点附近的虚假振荡。节点控制体中定义的流体速度多项式表达式也是从重构过程中得到的，用于计算相应网格顶点的移动速度。通过这种方式，网格顶点以严格的拉格朗日方式移动，尽管相邻节点控制体积之间仍然存在质量通量。因此该方案被称为拟拉格朗日方案。该方案对于质量、动量和总能量是保守的。进行了一些数值试验以证明该方案的准确性和鲁棒性。它对不连续问题具有良好的定性行为，对平滑问题具有最佳收敛速度。尽管相邻节点控制体积之间仍然存在质量通量。因此该方案被称为拟拉格朗日方案。该方案对于质量、动量和总能量是保守的。进行了一些数值试验以证明该方案的准确性和鲁棒性。它对不连续问题具有良好的定性行为，对平滑问题具有最佳收敛速度。尽管相邻节点控制体积之间仍然存在质量通量。因此该方案被称为拟拉格朗日方案。该方案对于质量、动量和总能量是保守的。进行了一些数值试验以证明该方案的准确性和鲁棒性。它对不连续问题具有良好的定性行为，对平滑问题具有最佳收敛速度。