A computational study of the spatial instability of planar Poiseuille flow is presented. A fourth-order finite difference with a fully implicit time-marching scheme is developed on a staggered grid. A semi-coarsening multigrid method is applied to accelerate convergence for the implicit scheme at each time step and a line distributive relaxation is developed as a fast solver, which is very robust and efficient for anisotropic grids. A new treatment for outflow boundary conditions makes the buffer area as short as one wavelength. The computational results demonstrate high accuracy in terms of agreement with linear theory and excellent efficiency in the sense that cost is comparable to (and usually less than) explicit schemes.