The implementation of a semi-implicit pressure-based fractional step method for compressible flows, and the incorporation of characteristic boundary conditions (NSCBC) is described. The discussion focuses on the discretisation of the pressure correction equation and the characteristic boundary conditions with respect to convergence rate of the algorithm and accuracy of the boundary conditions. An implicit implementation of the NSCBC is shown which removes the acoustic CFL limitation at boundaries while retaining the accuracy of the NSCBC approach. In this context, a new extrapolation method based on the Green-Gauss approach is proposed.