The compressible Euler equations can be used to describe the complex fluid flow problems such as implosion dynamics, ICF, interfacial instability and so on. These problems have the characteristics of dynamic boundary, strong discontinuity, large deformation, nonlinear, multimedia and so on. It is very difficult and challenging to solve these problems numerically. We study using Arbitrary Lagrangian-Eulerian（ALE） discontinuous Galerkin（DG） method for compressible Euler equations and construct a grid containing velocity ALE scheme, namely when the grid velocity is equal to the fluid velocity, It is the Lagrangian scheme; when the grid velocity is equal to zero, it is the Eulerian scheme. In one dimensional case, the speed of the grid is based on the Roe averaging method, which is characterized by the ability to accurately track the material interface. In the two-dimensional case, the speed of the grid is based on the method of solving the elliptic equation, and we have made a correction to it. The advantage of the scheme is that the new grid can be solved explicitly through computing physical quantities at the last time, avoiding the format iteration and interpolation process, and the grid distribution is more intensive on neighborhood of contact discontinuities and shock waves which it has some kind of grid adaptive function. Compared with the Lagrangian scheme, because grid speed is introduced into the scheme, we can avoid stopping calculation of pure Lagrangian scheme due to grid distortion. The numerical flux of the scheme is selected the HLLC flux. And to eliminate physical quantity non physical numerical oscillation by using discontinuous Galerkin method, the TVB slope limiter or WENO reconstruction is selected. The scheme can maintain the conservation of mass, momentum and total energy. Some numerical examples of single medium fluid flow and multimedia fluid flow show that the scheme has high accuracy, robustness and essential non oscillation.