  In this paper, we present the Stroboscopic Averaging Method (SAM), recently
introduced in [7,8,10,12], which aims at numerically solving highly-oscillatory
differential equations. More specifically, we first apply SAM to the
Schr\"odinger equation on the 1-dimensional torus and on the real line with
harmonic potential, with the aim of assessing its efficiency: as compared to
the well-established standard splitting schemes, the stiffer the problem is,
the larger the speed-up grows (up to a factor 100 in our tests). The geometric
properties of SAM are also explored: on very long time intervals, symmetric
implementations of the method show a very good preservation of the mass
invariant and of the energy. In a second series of experiments on 2-dimensional
equations, we demonstrate the ability of SAM to capture qualitatively the
long-time evolution of the solution (without spurring high oscillations).
