A Cartesian grid method has been developed for solving the 2D Euler and Navier-Stokes equations for viscous and inviscid compressible flow, respectively. Both steady and unsteady flows have been considered. Using a simplified ghost point treatment, we consider the closest grid points as mirror points of the ghost points. Wall boundary conditions are imposed at the ghost points of the immersed boundary. The accuracy of the method has been investigated for various test cases. We show computed examples of supersonic flow past a diamond-wedge airfoil and compare with analytical results. Further we compute time accurate solutions of the compressible Euler equations for an incident shock over a cylinder and compare the pressure time history with other work. The supersonic viscous flow around a NACA0012 airfoil is computed, and the lift and drag coefficients along with the pressure coefficient profile are compared with the literature. The method is also tested for supersonic flow over a cylinder, and the computed skin friction profiles have been used to assess the accuracy. Lastly the supersonic flow around a 2D F-22 fighter aircraft with simulated jet engine outflow is shown to illustrate the flexibility of the method. The present method is built on a previously established simplified ghost point treatment, but performs better. The results are comparable, although not as accurate as other more complex methods.