The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step has been based on CFL conditions established for fixed and uniform meshes. This work provides a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the $\alpha$-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the $\alpha$-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers' and Euler equations are presented.

中文翻译：

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自适应移动网格守恒定律DG解的CFL条件研究

在通常采用显式步进的自适应移动网格上双曲守恒律的不连续伽辽金 (DG) 解中，时间步长的选择在提高稳定性和效率方面起着至关重要的作用。常用的时间步长选择基于为固定和均匀网格建立的 CFL 条件。这项工作为实际自适应 DG 计算中使用的那些时间步长选择策略提供了数学依据。对线性标量守恒定律的移动网格 DG 方法进行稳定性分析。在此分析的基础上，提出了一种新的时间步长选择策略，它考虑了 $\alpha$ 函数（与通量和网格运动速度的雅可比矩阵的特征值有关）和网格元素的高度的耦合。分析还提出了在数值方案和时间步长选择中 $\alpha$ 函数选择的几种稳定组合。给出了用移动网格 DG 方法获得的 Burgers 和 Euler 方程的数值结果。