||A Unified Coordinate approach in describing the fluid motion for computational fluid dynamics has been introduced by Hui et. al. (W.H. Hui, P.Y. Li and Z.W. Li, A Unified Coordinate System for Solving the Two-Dimensional Euler Equations, J. Comp. Phys. 153 (1999), 596-637). This coordinate system moves with velocity hq where h is arbitrary and q is the velocity of the fluid particle. Hence, it includes the Eulerian Coordinates and the Lagrangian Coordinates as two particular cases where h ≡ 0 and h ≡ 1 respectively. h is suggested to be chosen so that the grid angle is preserved. One of the purposes of this thesis is to explore the possibilities on choosing this free function h. Improvements on the accuracy and the efficiency of the h-equation solver will be given. We will also discuss the way to improve the efficiency of the computations using the Unified Coordinates, including the replacement of the Exact Riemann Solver and the introduction of the Composite Operator approach.