In this paper we develop the Runge-Kutta discontinuous Galerkin finite element method for two-dimensional compressible gas dynamic equations in unified coordinate. The equations for fluid dynamics and geometry conservation laws are solved simultaneously. All the calculations can be performed on the fixed meshes. The information about the grid velocities is not needed in calculation. Some numerical examples are given to evaluate the efficiency and the reliability of the scheme. The numerical results show that the algorithm works well.