In this paper, the preconditioned multistage time stepping methods which are the popular multigrid smoothers are studied for the compressible flow calculations. Fourier analysis on the local time stepping and block-Jacobi preconditioned residual operators is performed using the linearized 2-D Navier-Stokes equations. It turned out that the block-Jacobi preconditioner has better performance in eigenvalue clustering. They are implemented in the 2-D compressible Euler and Navier-Stokes calculations with multigrid methods to verify that the block-Jacobi preconditioned multistage time stepping shows better performance in convergence.