  We build Gaussian wave packets for the linear Schr\"odinger equation and its
finite difference space semi-discretization and illustrate the lack of uniform
dispersive properties of the numerical solutions as established in Ignat,
Zuazua, Numerical dispersive schemes for the nonlinear Schr\"odinger equation,
SIAM. J. Numer. Anal., 47(2) (2009), 1366-1390. It is by now well known that
bigrid algorithms provide filtering mechanisms allowing to recover the
uniformity of the dispersive properties as the mesh size goes to zero. We
analyze and illustrate numerically how these high frequency wave packets split
and propagate under these bigrid filtering mechanisms, depending on how the
fine grid/coarse grid filtering is implemented.
