In this report, we propose a collection of methods to make such an approach possible for Euler equations in one and two dimensions. We propose an explicit single-step ALE DG scheme for hyperbolic conservation laws. The scheme considerably reduces the numerical dissipations introduced by the Riemann solvers. We show that the scheme also preserves the constant states for any mesh motion. We then study the effect of mesh quality on the accuracy of the simulations, and based on that, come up with a mesh quality indicator for the ALE DG method. Based on the considerations from the study on mesh quality, we design a local mesh velocity algorithm to compute the motion of the mesh. And finally, we propose a local mesh adaptation algorithm to control the quality of the mesh, and prevent the mesh from degradation.

翻译：本报告提出一系列方法，使得该途径能够适用于一维和二维欧拉方程。我们针对双曲守恒律提出一种显式单步ALE DG格式。该格式显著降低了黎曼求解器引入的数值耗散。我们证明该格式在任何网格运动下均能保持常值状态。随后研究网格质量对模拟精度的影响，并以此为基础推导出ALE DG方法的网格质量指示符。基于网格质量研究的考量，我们设计了一种局部网格速度算法来计算网格运动。最后，我们提出一种局部网格自适应算法，用于控制网格质量并防止网格退化。