  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.
